Revised the fifth time about 11:17 PM, EST, saturdayy 03-20-2021.
The evidence I found indicates that the record age to beat for the youngest monarch to become a father is 12 years, though there are possibilities of younger ones.
Part One of Twelve: Some of the Problems and Difficulties:
That is a difficult question to answer because in some regions monarchy may have existed for thousands of years before recorded history, and because the ages of many monarchs and their offspring are not known precisely, and because the younger the offiical father was, the more likely people would be to suspect that he wasn't really the father.
For example co King Sigurd II Munn of Norway was the alleged father of several claimants and kings of Norway who were believed to be his illegitimate sons.
After Kings Eystein II and Sigurd II (killed 6 February 1115) were killed by followers of King Inge in 1157, the followers of the slain kings selected ten-year-old Haakon, (1147-1162), an illegitiamte son of Sigurd II, to be their king. Presumably Haakon II was the oldest illegitimate son of Sigurd II that they knew about. Sigurd Munn is said in many sources, such as Wikipedia, to have been born in 1133, thus making him 13 or 14 when Haakon II was born.
So King Sigurd III Sigurdsson Markusfostre who was selected as king by the followers of Haakon II after he was killed in 1162 was probably younger than Haakon, and so older than 7 years and 7 months, even if born posthumously, and probably less than 16 when he was beheaded by followers of rival king Magnus V on September 29, 1163.
Sverre arrived in Norway in 1176, claiming to be an illegitimate son of Sigurd Munn, and after civil war, became sole king in 1184.
The tale told in Sverre's saga is the official version. Historians have questioned the veracity of it, especially with regard to Sverre's alleged paternity.2 Some historians have considered his claim to be King Sigurd's son to be false, as did many of his contemporaries. Others have believed the paternal claim to be true, while most historians have found that the paternal question cannot be given a definite answer.2 Although the fact that kings fathered illegitimate sons was taken for granted, other facts indicate that Sverre was in his early thirties when he came to Norway, such as the age of his own sons and nephews. It has been cited against Sverre's claim that according to Canon law, one had to be at least 30 years old to be eligible for the priesthood. If Sverre was 30 years old when he became a priest, this would place his birth no later than 1145, making his paternal claim impossible, as Sigurd Munn was born in 1133. This particular objection has lost credence as it has become clear that this age limit was routinely ignored in Scandinavia at the time. However, other objections remain, such as the fact that Sverre consistently refused to undergo an ordeal by fire to prove his claims. At the time, such a trial was routine for new claimants to the throne, and belief in its efficacy seems to have been universal; yet Sverre refused to undergo it.
Another factor in favor of Sverre's claim is that the site Medieval Lands, in the page about King Harold Magnusson, says about his son King Sigurd II:
Snorre names Sigurd as the son of Harald by "Thora, a daughter of Guthorm Grabarde", implying that he was born before his father's accession and was older than his half-brothers. He succeeded in 1136 as SIGURD Joint King of Norway.
So if Sigurd II was born before his father succeeded in 1130 he would have been 17 or older when Haakon II, and presumably his other reported children, were born, thus making his paternity more probable and also less noteworthy.
And this is an example of sone of the problems with answering such a question with information from eras which have poor records.
But I will add to this answer later with better recorded and more probable examples of young royal fathers.
One way to search for young royal fathers is to find examples of many generations of a royal family who die in a comparativel short period of time.
For example, part of the genealogy of the dynasty of gWynedd in Wales is:
Cadwalladr the Blessed (died 662 or 688), son of Cadwallon (died 633), son of Cadfan, son of Iago, son of Beli, son of Rhun, son of Maelgwyn Gwynedd (died 547/548/549), son of Cadwallon Lawhir, son of Einion, son of Cunedda.
Despite the fact that Welsh genealogies are more likely to be accurate than Welsh chonology, it has been argued that the large number of generations who died between Maelgwyn and the second Cadwallon - 6 generations & 5 generation gaps in 84 to 86 years - is evidence that the genealogy is inaccurate. To make it seem even less plausible, Cadwallon Lawhir has been said to have died in 533, making 7Generations and 6 genration gaps dying in 100 years - even though that entry is obviously that of the second Cadwallon moved a century earlier.
So I began collecting examples of royal lineage where the same number of geratins died in similar periods of years. IN some cases that is because the king in the first generation died much older than the king in the last gerneraton, in others because the kings became fathers at very young ages, in many both.
Examples of such lineages are in the Northern Wei, Tang, and Ming dynasties of China, the Yamato dynasty of Japan, the Joseon Dynasty of Korea, The Qajar Dynasty of Iran, the Nawabs of Bahawalphur in Pakistan, the Abbassid Caliphs, etc.
And in some cases the large number of generations in a short time is due to a very young average age of fatherhood.
Part Two: Chinese Emperors Who Became Young fathers:
Here is a link to family tree of emperors of the Tang Dynasty of China.
Note that the eight generations and seven gaps between generations from Li Heng - Emperor Suzong - born 711 to Crown Prince Li Pu born in 824 there are 113 years, for an average of 16.14 years per generation gap.
Li Heng - Emperor Muzong - was born July 26,795, and his son Li Zhan - Emperor Jingzong - was born July 22, 809, which makes Muzong 13 years, 11 months, and 26 days older than Jingzong.
And take a look at this family tree of the Wei Dynasty of China:
The nine generations and eight generation gaps from Tuoba Gui - Emperor Daowu - born 371 to Yuan Xu - Emperor Xiaoming - born 510, cover 139 years, making an average of 17.375 years between generations.
The five generations and four generation gaps between Crown Prince Tuoba Huang, born 428, and Yuan Ke - Emperor Xuanwu - born 483, cover 55 years, averaging 13.75 years per generation gap.
Touba Hong - Emperor Xianwen - was born in 454, only 13 years before his son Tuoba Hong - Emmeror Xiaowen - was born in 467.
Crown Prince Tuoba Huang was born in 428, only 12 years before his son Tuoba Jun - Emperor Wencheng - was born in 440. Depending on when during the years 428 and 440 he and his son were born, Tuoba Huang could have still been 11 when Wencheng was born.
If you can find a copy of Arthur Christpher Moule, The rulers of China, 221 B.C.-A.D. 1949; chronological tables. With an introductory section on the earlier rulers c. 2100-249 B.C., Routledge and K. Paul, 1957, B0007JWGTC, you will find that it gives precise birth dates for most of the Emperors in those dynasties, enabling more precise ages of fatherhood to be calculated.
Part Three: An Ottoman Example:
The Ottoman Sultan Mehmed II was born 30 March 1432, and his son Sultan Bayezid II was born in December 1447, 14 years and almost 9 months later.
I note that A. D. Alderson's book on the Ottoman genealogy, The Structure of the Ottoman Dynasty, 1956, 1982 mentions an Ottoman prince, Alaeddin Ali, a son of Murad II, who was killed in 1443. It gives him a birthdate making him 13 in 1443, as well as two sons, and he would have been 12 when the older son was born.
I have often wondered whether there was a typographical error in the birthdate of Alaeddin Ali, and he was actually older. He was killed 39 or 40 years after hs father's birth, and so could have been over 20 when he was kiled.
According to the article on Murad II in Wikipedia, Alaeddin Ali was born in 1425, and so he could have been about 16 to 18 when his sons were born if they were born at the dates given by Alderson.
So clearly more reserch on the biography of Alaeddin Ali would be necessary for any certainty. In any case, he never became Sultan.
Part Four: A Japanese Example:
Tsunehito - Emperor Kameyama - was born 9 July 1249 and his oldest child, Princess Kenshi, was born in 1262, when he was 12 or 13 years old.
An English language source for the genealogy of the Japanese imperial family is Richard Arthur Brabazon Ponsonby-Fane, The Imperial House of Japan, 1959. It might have a precise birthdate for Princess Kenshi, enabling Kamayama's age at her birth to be calculated.
Part Five: An Ethiopian King of Kings:
Ethiopian King of Kings Yaqob I was born about 1590 and reigned from 1597 to 1603 and again from 1604 to 1606, and was killed at the Battle of Gol 10 March 1606. Yaqob I had several sons.
Yaqob had married some years before a foreigner named Nazarena, by whom he had three sons, one of whom had died before the Battle of Gol. Nazarena sent her surviving sons to safety in exile: Cosmas, the older, went south and was not heard of again; the younger, Saga Krestos, went to the safety of the Kingdom of Sennar where he was treated well and came of age. When King Rabat proposed that Saga Krestos marry his daughter, Saga Krestos refused, and was forced to flee to another refuge, adopting Roman Catholicism while at Jerusalem. Eventually he found his way to Rome (1632), and eventually to Paris, where he was given lodgings by Cardinal Richelieu. Saga Krestos died of pleurisy in 1638 at the age of 38. Thomas Pakenham provides a brief sketch of Saga Krestos' European life in his The Mountains of Rasselas, and the book ends with a description of Pakenham's visit to Saga Krestos' grave in Rueil-Malmaison.2
However, O. G. S. Crawford has cast doubts on this story. In an article that discusses the surviving sources for the story of Saga Krestos, he points out a number of problems in his story which include a discrepancy over the possible date of his birth (i.e., Saga Krestos is likely to have been born in either 1610 or 1616, whereas Yaqob died in 1607), and the story of three Ethiopian monks who report that Saga Krestos was an apostate monk who wandered from place to place begging for money.3
Thus it seems uncertain whether Yaqob I is a candidate for the youngest royal father.
Part Six: An Inca Example:
There is also some confusion about the age of fatherhood of the Inka Emperor Huayna Capac or Wayna Qhapaq. Wikipedia says he was born about 1464-1468 and died in 1524.
It says that his sons Athualpa or Atawallpa and Huascar or Waskar, who fought an Inkan civil war, were born about 1502 and 1503 respectively
However, I once read an account that claimed that Huayna Capec was about 13 years older than Athualpa and Huascar, which would make him born about 1488 to 1490 according to their Wikipedia birth dates.
And that source also claimed that Huayna Capec was only about ten years older than his heir Ninian Cuyochi, which would make Ninian Cuyochi born about 1498 to 1500 if Athualpa and Huascar were born when Wikipedia says they were. The Wikipedia article on Ninian Cuyochi says he was born in 1490, which would make Huayna Capec born about 1480 if he was only about ten years older than Ninian Cuyochi.
I note that the Wikipedia article on Huayna Capec says:
Huayna Capac[note 1] (1464/1468–1524) was the third Sapan Inka of the Inca Empire, born in Tumipampa10 sixth of the Hanan dynasty, and eleventh of the Inca civilization.
And also says:
Born c. 1468
Tumebamba or Cusco, Peru
The exact place and date of Wayna Qhapaq's birth are unknown. Though he was raised in Cusco, he may have been born in 1468 in Tumebamba (modern Cuenca) and have spent part of his childhood there.1517 He was the son of Thupaq Inka (ruled 1471-1493) who had extended Inca rule north into present-day Ecuador, a process continued by Wayna Qhapaq.19
As a "boy chief" or "boy sovereign", Wayna Qhapaq had a tutor, Wallpaya,9:218 a nephew of Inka Yupanki. This tutor's plot to assume the Incaship was discovered by his uncle, the Governor Waman Achachi, who had Wallpaya killed.14:109
If Huayna Capec was born in 1464-1468, and if his father died in 1493, Huayna cApec would have been aged 25 to 29 when he became Inca, and I find it hard to believe that someone that old would be considered a boy ruler.
Thus I consider the story in one source that Huayna Capec could have become a father as young as ten to be possible but not proved.
So it seems that the record to beat for youngest monarch to become a father is 12 years old.
Added 03-18-2021 European examples.
I don't think that there are any proved European examples of royal fathers as young as the Chinese and Japanese examples. But I will try to list some of the youngest European royal fathers.
Part Seven: Edward III:
In Britain, King Edward III of England was born 13 November 1312, and his oldest son Edward, Prince of Wales, was born 17 years, 9 months, and 2 days later on 15 June 1330.
This article claims he ws the youngest British monarch to become a father:
Part Eight: Louis XV:
King Louis XV of France was born 15 February 1710 and died in 1774. his fourth child and oldest son Louis, Dauphin of France was born 4 september 1729, when his father was 19 years, six months, and 20 days old. But Loiuis XV was only 17 years, 5 months, and 29 days old when his oldest daughter Louise Elisabeth was born on 14 Augusut 1727.
Part Nine: Osman II:
If he counts as a European monarch, Ottoman Padishah Osman II, with his capital in Constantinople, was born 3 November 1604 and his son Shezade Omer was born 16 years, 11 months, and 14 days later on 20 Ocotber 1621.
Part Ten: Emperor Frederick II:
Emperor Frederick II, King of Jerusalem and Sicily, was born 26 December 1194, and his oldest son, Henry VII, King of the Romans and King of Sicily, was born in 1211, and so when Frederick was very probably sixteen years old - somewhere between 16 years and 5 days and 17 years and 4 days old.
Part Eleven: Emperor John V Palaiologos:
Emperor John V Palaiologos was born on 18 June 1332, and his oldest child Emperor Andronikos IV was born 15 years, 9 months, and 15 days later on 2 April 1348.
But the article on Andronikos IV says he was born 9 days later on 11 April 1348, as does Medieval Lands, making JOhn V 15 years, 9 months, and 24 days old:
Part twelve: Eric II of Norway:
Mentioned in the answer by Lars Bosteen, Eric II of Norway was born in 1268, and became the father of his older daughter, Margaret the Maid of Norway, acknowledge das the rightful Queen of scotland, by 9 April 1283.
If Eric was born sometime between 1 January 1268 and 31 December 1268, and Margaret was born on 9 April 1283, Eric would have been between 14 years, 3 months, and 9 days, and 15 years, 3 months, and 8 days, old when Margaret was born. If Margaret was born before 9 April 1283 Eric's possible age range would be correspondingly younger.
According to Wikipedia Erick was born in 1268 and Maargaret was born in March or up to 9 April 1283. That would make Eric from 14 years and 2 months old to 15 years, 3 months & 8 days old when Margaret was born.
Part Thirteen: Sultan Ahmed I:
The Ottoman Sultan and Padishah Ahmed I, if he counts as a European monarch, was born in April 1590 and his oldest child, Padishah Osman II, was born 3 November 1604, when Ahmed I was 14 years and about 6 to 7 months old.
Part Fourteen: Merovingians:
The Frankish Kings of the Merovingian dynasty had a number of teenage, and possibly some pre teen, fathers.
King Clovis I of the Franks had an oldest son KIng Theoderich or Theuderic I of Austrasia, who was born about 485 or about 487 and had an oldest son King Theodebert or Theudebert I of Austasia, who was born about 499 to 504, or about 503. So Theuderic I was about 14 to 19 years, or about 16 years, older than his son King Theudebert I, depending on the source.
King Chlothachar/Clothair/Lothar I was born about 501 or 502 and had an oldest son Gunthar who was born about 517 and thus when his father was about 14 to 16.
But Wikipedia says that Clothar I was born about 497 and thus should have been about 20 when Gunther was born.
King Gunthramn or Gontran or Guntram was born about 532/34 and had an illegitamate son Gundobald born about 549 when his fther was about 15 to 17.
King Childebert II was born about 570 and his sons KIng Theodebert II and KIng Theoderich II were born about 586 and 587, when Childebert II was about 16 and 17.
King Chilperic I was born about 535 and so was about 13 to 16 when his oldest child Theodebert was born about 548/51.
Wikipedia says that Chilperic I was born about 539, which would make him about 9 to 12 when Theodebert was born, but Wikipedia doesn't give birthdates for the oldest children of Chilperic I.
King Charibert II may have been born about 618/619, and died in 631. He was survived by a son Chilperic who might have been born about 630 and died young after 631. That makes Charibert II about 11 to 12 years old when Chilperic was born.
a) [CHILPERICH (-after 631 young). Fredegar records that "Charibertus rex" left "filium parvolum…Chilpericum" who only survived his father for a short time, recording that "it was said that Dagobert's followers brought about his assassination". If the marriage date of Chilperich's paternal grandparents is correct as shown above, Chilperich's father could not have been more than twelve years old when his son was born.]
But Wikipedia says that Charibert II died in 632 and was born about 607 to 617 and then says:
There are no direct statements about when Charibert was born exactly, the only known fact being that he was "a few years younger" than his half-brother Dagobert.1 His father Clotaire evidently had a bigamous marriage (not then uncommon) and he was the offspring of the junior wife.
So according to Wikipedia Charibert might have been about 13 to 25 when his son Chilperic was born.
King Clovis II was born 633 and his oldest sons Clothaire III and Theoderich II were born about 650 and 651 respectively, making him a father at about the ages of 16 to 17 and 17 to 18.
But Wikipedia says that Clovis II's oldest son Clothaire III was born in 652, making Clovis II 18 or 19 at the time.
King Childerich II, the third son of Clovis II, was murdered sometime between 18 October & 10 November in 675 along with one of his sons, Dagobert, who was obviously very young.
According to Wikipedia, Childerich II was born in 653 and his son Dagobert was five years old in 675, Thus making Childrich II about 17 or 18 years older than Dagobert.
Clovis III or IV, older son of Theoderich III, was born about 678 and died in 695. His younger brother Childebert III was king from 695 to 711. Childebert III's son Dagobert III was born about 697/698 which makes Childbert III no more than 16 to 18 when his son was born, possibly much younger. Dagobert III's son Theoderich IV was born about 712, which was about 34 years after his grandfather's older brother was born, making an average of no more than 17 years per generation, and making Dagobert III about 14 or 15 when Theoderich IV was born.
Wikipedia says about Clovis IV:
He was born around 6773 or possibly towards 682.4
Wikpedia says that Childebert III was born about 678/679.
And that his son Dagobert IV was born about 699.
And that Dagobert IV's son Theuderic IV was born in 712, and thus when Dagobert IV was about 12 or 13.
Added 03-202-2021 More Minor age Merovingian Monarchs.
I found a few more minor age Merovingian monarchs who became fathers in the family tree of King Sigebert I (about 535?-575) and his queen Brunechildis or Brunhilda (about 545/50-613).The had three daughters and one son King Childebert II (about 570-596).
Childebert II had two sons, King Theodebert II (about 586-612) and King Theoderich II (587-613). KIng Theodebert II had a son Merovech or Mereveus who was born about 611 and killed in 612. King Theoderich II had four sons; King Sigebert II (about 602/603-613), Childebert (about 603/604-?0), Corbus (about 604/605-613), and Merovech (about 607/608-?).
So according to Medieval Lands Childbert II was about 16 or 17 years older than Theoderich II and about 15 or 16 years older than Theodebert II. King Theoderich II was about 20 to 21 years older than his son Merovech, about 17 to 18 years older than his son Corbus, about 16 to 17 years older than his son Childebert, and about 15 to 16 years older than his son King Sigebert II.
Wikepedia says that Childebert II was born about 570 and his sons Theudebert II and Theuderic II were born about 585 and in 587, Thus making Childebert II about 15 and 17 yers older than them. It also says that Theuderic's son, Merovech, Corbo, childebert, and Sigebert II were born in 604, 603, 602, and 601 repectively, making Theuderic turn 17, 16, 15, and 14 in the years they were born.
And thus it is seen that the youngest European King to become a father was probably a Merovingian, though it looks like a lot of research would be needed to determne who it was and how old he was.
Added 03-29-2021. In part five I discuss the possibility that Ethiopian King of Kings YaqobI might have become a father at a very young age.
Here is a link to an article claiming that his alleged son Sagga Krostos was not his son.