Part One: It is Partially Confirmed.
I Found the 22nd canon of the Council of Tours of 567:
- Some still hold fast the old error, that they should honour the 1st of January. Others, on the festival of the See of Peter, present meat offerings to the dead, and partake of meats which have been offered to demons. Others reverence certain rocks, or trees, or fountains, etc. The priests should root out these heathenish superstitions.
I don't know if the "old error" of "honoring" the 1st of January means using it as the beginning of the year.
I also don't now how binding, if any, the 22nd canon of the Council of Tours in 567 was on Catholics outside of Gaul or for how long. I don't know how influential it was in the process of adopting Christian Holy days as the beginning of the year, even when those holy days were inside a month and resulted in splitting that month between two calendar years.
I suspect that a site about church history might have more information on the subject.
Part Two: Corrections to the Original Question.
The original question has inaccurate data about the history of the dates used for the beginning of the year in Europe.
it wasn't until the reforms in 1752 that January 1st as New Year's day became widespread (i.e., the protestants didn't accept Pope Gregory's calendar until then. That is why George Washington had two different birthdays).
The Roman Republic used different dates to start the one year terms of the Consuls. In 153 BC the consuls began their term on January 1, which remained the date for many centuries into the era of the empire.
January 1 was the start of the year in the Julian Calendar adopted in 45 BC. There were many local calendars used in the Roman Empire, and even when localities adopted the Julian calendar, they sometimes used different days as the start of the year.
They also used various calendar eras to count the years from.
The Roman calendar began the year on 1 January, and this remained the start of the year after the Julian reform. However, even after local calendars were aligned to the Julian calendar, they started the new year on different dates. The Alexandrian calendar in Egypt started on 29 August (30 August after an Alexandrian leap year). Several local provincial calendars were aligned to start on the birthday of Augustus, 23 September. The indiction caused the Byzantine year, which used the Julian calendar, to begin on 1 September; this date is still used in the Eastern Orthodox Church for the beginning of the liturgical year. When the Julian calendar was adopted in AD 988 by Vladimir I of Kiev, the year was numbered Anno Mundi 6496, beginning on 1 March, six months after the start of the Byzantine Anno Mundi year with the same number. In 1492 (AM 7000), Ivan III, according to church tradition, realigned the start of the year to 1 September, so that AM 7000 only lasted for six months in Russia, from 1 March to 31 August 1492.
During the Middle Ages 1 January retained the name New Year's Day (or an equivalent name) in all western European countries (affiliated with the Roman Catholic Church), since the medieval calendar continued to display the months from January to December (in twelve columns containing 28 to 31 days each), just as the Romans had. However, most of those countries began their numbered year on 25 December (the Nativity of Jesus), 25 March (the Incarnation of Jesus), or even Easter, as in France (see the Liturgical year article for more details).
So far today I have been unable to find a source saying that the Grand Duchy of Lithuania adopted January 1 as the beginning of the year in 1362.
Most western European countries shifted the first day of their numbered year to 1 January while they were still using the Julian calendar, before they adopted the Gregorian calendar, many during the 16th century. The following table shows the years in which various countries adopted 1 January as the start of the year. Eastern European countries, with populations showing allegiance to the Orthodox Church, began the year on 1 September from about 988. The Rumi calendar used in the Ottoman Empire began the civil year on 1 March until 1918.
The table shows that during the 16th century there was a movement for some reason to make January 1 the beginning of the year. The table lists 10 states which adopted January 1 as the beginning of the year before the Gregorian calendar of 1582, starting with the Holy Roman Empire in 1544.
So the majority of Catholic and Protestant Europeans adopted January 1 as the beginning of the year before Great Britain did so in 1752, and the majority of Roman Catholic Europeans did so before the Gregorian Calendar was started in 1582.