Why isn't the solstice New Years Day?

Merry Christmas all anyway.
The original Roman calendar started with March. That is why our 9th, 10th, 11th, and 12th months are named after the Latin words for 7, 8, 9, &10 respectively.
If, as Nonukes found, the setting of 25th Dec for Jesus's birth was in the 4th century, the drift of the solstice meant that would probably have been the date for it then.

dwise1 shows that I went the wrong way in moving the solstice.
I should be more sure of that on which I pontificate!

Nothing wrong with your posting from Wiki. In the 16th century 11 days were dropped from the month of October to bring it back in line. The days of the week were unaffected. The original Julian calendar had about 3 leap years too many every 400 years. With the Gregorian revision I think it is some tens of thousands of years before we will need one more or less leap year to keep things in step. Maybe they should have dropped a couple more days in 16th century to get the solstice back to the 25th!
Not all countries adopted the change at the same time - it was 18th century for England.

The Wikipedia article on the Gregorian calendar states that in the 4th century the solstice was regarded as 21st in the Alexandrian church, and 25th in the Roman, without specifying any reason for the difference. The Alexandrian view eventually prevailed. I haven't checked when it became associated with Jesus's birth.

Adding to what NoNukes has said, the rotation of the Earth slows by a few milliseconds per day per century, and has gained that 0.002 sec since the second was defined. This continuous slowing has to be taken into account in determining the local time for astronomical events in the past, as it adds up to a matter of hours BC. If civilisation survives, and the second is not redefined, our distant descendants will need a leap second every day!

In round figures, we currently need a leap second every 18 months. At current rates of slowing, in 100 years there will be one needed every 9 months, in 200 every 6 months, 1000 years about every 2 months, 10,000 years every week, 50,000 to 100,000 years it will reach every day. Our vastly superior descendants will work out an answer. (Or our post-civilisation cave-dwellers won't care!)