Why is January 1st New Year's Day and not, say, the winter solstice or the vernal equinox? Good question!
The simple answer is that this day was established by the Roman emperor Julius Caesar (the Julian Calendar) in the year 46 BC and was adopted and decreed for the entire Christian world by Pope Gregory XIII (the present Gregorian Calendar) in the year 1582 AD. Having said that, we are still left with the question why Julius Caesar (or the calendar which he used) chose their first of the month and did not rather align it with the winter solstice, for example.
An earlier Roman calendar (before the Julian) actually began the year with the vernal equinox, the date for which Julius set at 25 March (which, because of inaccuracy of the Roman calendar, ends up being our 21 March).
It is an interesting fact that the day of the Incarnation for the Christian world is the 25 of March, the exact day of the vernal equinox established by the Julian Calendar and the actual New Year's Day of the more ancient Roman calendar, and that the numeration of the years in our present calendar has the Incarnation of Christ our Lord as the reference point: viz. B.C. and A.D. Given all of this it remains indisputable that New Year's Day is a Christmas celebration in the present day calendar established by the Pope. It is the Octave Day of Christmas: the day of the Circumcision and naming of the Lord "Jesus" according to Sacred Scripture. (N.B. The Gospel for today's New Year's Day Mass: Luke 2:21 "And when eight days were fulfilled for his circumcision, his name was called Jesus, the name given him by the angel before he was conceived in the womb.")
P.S. As I mentioned in a previous post about the Bethlehem Star in Jesus of Nazareth: Infancy Narratives it is surprising that Pope Benedict does not even address the question of the dates related to Christmas in that book (though he does mention the question of the year of the Incarnation as the reference point for our calendar).