As the title says, has any person who has held or has gone on to hold the title of King (or some other similar monarchical title) ever been a serving admiral, who has exercised actual command of a fleet at sea, and not just an honorary admiral?
The king/admiral in question need not have ever commanded a fleet in battle or even during time of war, just that they have held the rank and performed the duties of an admiral at some point during their life.
Please provide examples if any exist.
It turns out William IV wasn't called the "Sailor King" for nothing. He was a younger son of George III and thus didn't expect ever to become king. His naval career began at the age of thirteen as a midshipman, in the late 1770s. He later served in NYC (British-occupied) during the Revolution, and George Washington actually plotted to kidnap him. He was promoted to rear admiral and to command of HMS Valiant in 1789. He left the naval service in 1790.
He was made a full admiral in 1798 and Admiral of the Fleet in 1811, but those positions were either honorary or administrative. However, the rear admiral rank he held in 1789-1790 was a real serving position, and not honorary at all.
William IV did not become king until 1830, his older brother George IV having died (and George IV's only legitimate child had predeceased him, as had the intervening brother in order of age).
In the honorable mention category would be Andrea Doria, one of the greatest admirals who ever lived, who was offered to become lord of Genoa (but turned it down and became "perpetual censor", in the ancient, not modern, sense of "censor", instead).
How stuck are you on the exact title of admiral? Surely many kings of Norway and Denmark (and some of England), like Sweyn Forkbeard and Cnut, in the 10th and 11th centuries were functionally admirals even if their concept of organization gave no equivalent to the modern notion of admiral. The North Sea Empire would hardly have been possible otherwise.
Finally do you count people who used command of a fleet in order to seize the crown, even if that wasn't their main occupation? That list gets a lot longer since it includes the Roman Emperor Heraclius, and of the English Kings at the very least William I, Louis I/VIII (arguably as to king of England -- but he later became king of France unarguably), Edward IV (second time), Henry VII, and William III. (Not listing the North Sea Empire kings as I already mentioned them.)
All in all I'm guessing William IV holds the record for longest gap between becoming admiral and becoming king, at over 40 years. (Louis I/VIII had a 7 year gap between his temporarily successful invasion of England and succeeding to the French throne. On the other hand, none of William IV's children have been canonized...that we know of.)
During the Greco-Persian Wars, King Leotychidas II of Sparta (reigned 491 to 476 BC) was appointed admiral in overall command of the Greek fleet in 479 BC, while he was king. He was in command at the time of the naval / land battle at Mykale.
This was the only time we know of that a Spartan king held this rank. Note that Leotychidas was appointed to this rank by the Spartan state; he did not have it by default simply because he was king.
The Eurypontid King Leotychidas II was the Spartan Navarch (ναυαρχος, admiral) for the year 479 BC during the Greco-Persian Wars. He was king at the time, having reigned since 491 BC. Sparta usually appointed a different admiral every year (though the rules were later bent somewhat due to the successes of Lysander) towards the end of the Peloponnesian War.
He was appointed in the aftermath of the Battle of Salamis, supposedly because the Spartans did not want him commanding the army, the position usually held by Spartan kings. This was because he had allegedly offended Apollo by bribing the Oracle at Delphi. Thus, instead of commanding the army at the Battle of Plataea in 479 BC, he led the combined Greek fleet at the Battle of Mycale. Leotychidas,
as commander-in-chief of the ‘Hellenic League’ fleet, ...fomented the revolt of *Chios and *Samos, and decisively defeated the Persians in a land and sea battle off Cape Mycale.
Source: Paul Cartledge, 'Leotychidas II'. In 'Oxford Classical Dictionary'
Leotychidas was exiled in 476 BC, this time accused of accepting a bribe. He died in exile, probably after 470 BC.
As @C Monsour has pointed out in his excellent answer, rulers in medieval times and earlier were often "functionally admirals" without having a specific title to that effect. Most simply wouldn't have felt the need to appoint themselves 'admiral' or 'commander of the navy'.
This is what makes the case of Leotychidas II different from the usual ancient practice, and thus worth mentioning; Sparta did have a title recognizable as 'admiral', 'commander of the navy' or 'commander of the fleet' (as opposed to land forces or other 'duties' of a ruler). Further, they later created the post of deputy navarch for Lysander as Spartan law did not allow someone to hold the position of navarch two years running. Even Athens, the main Greek naval power for most of the Classical period, did not have a 'navarch'; instead, they used the title 'strategos' (general) (i.e. the same as land forces).
That said, for ancient times, you may also want to consider Ramses III of Egypt and his defeat of the Sea Peoples at the Battle of the Delta (circa. 1175 BC) as he does appear to have personally commanded (but I'm not aware of any 'admiral' or equivalent title, and the surviving inscription accounts may well credit the pharaoh with more than his due).
Other source:
Alfred S. Bradford, 'Leonidas and the Kings of Sparta' (2011)
King James II was commander of the Royal Navy from 1660 to 1674, during the second and third Anglo-Dutch wars, before he became king. The text accompanying the image of his portrait at the Maritime Museum says
He commanded the fleet that routed the Dutch at the Battle of Lowestoft in 1665, the first action of the Second Dutch War, and with less success at the indecisive action at Solebay in 1672. This was the last time that a prince and future monarch directed a fleet in action, and the occasion when his flag-captain, Sir John Narborough recorded his belief that 'no prince upon the whole earth can compare with His Royal Highness in gallant resolution in fighting his enemy...he is general, soldier, pilot, master, seaman [and] to say all , he is everything that a man can be, and most pleasant when the great shot are thundering about his ears.'
He fought at the Battle of Lowestoft in 1665 and the Battle of Solebay in 1672.
Lars Bosteen mentioned a Spartan King and Admiral in the Greco-Persian wars.
Artesmisia I was queen of the city state of Halicarnassus and the nearby islands of Kos, Nisyros, and Kalymnos. Her kingdom was part of the Persian Empire's Satrapy of Caria. Queen Artemisia commanded a fleet of 5 galleys as a contingent in the Persian fleet in Xerxes's invasion of Greece. She commanded her fleet at the naval battles of Artemisium and Salamis in 480 BC.
Here is a link to a similar question and its answers
There are a number of examples of Roman fleets being commanded by consuls, who were the Roman equivalent of the earlier Etruscan kings.
The second and third Punic Wars featured less naval action.
