WE DON'T CELEBRATE the civil New Years' Day for several reasons.
We already have four Jewish New Year celebrations.
1. First of Nisan is the New Year for (Jewish) kings and for the religious calendar (for festivals)
2. First of Elul is the New Year for the tithing of cattle (but see 3., the first of Tishri)
3. First of Tishri is the New Year for the civil calendar (including the counting of the reigns of foreign kings)
4. First of Shevat is the New Year for trees, according to Bet Shammai, but Bet Hillel fixed the date as the 15th of Shevat
The sources for the holidays are given in the Jewish Virtual Library.
But the followers of Jesus are, perhaps unwittingly, celebrating a Jewish occasion.
Check the math.
Assuming Jesus was born on December 25 - which most historians reject - then January 1 is eight - count'em, eight - days later.
Almost every Jewish boy knows what happened on his eight day (if he was healthy). Two words: BRIT MELAH, circumcision, hopefully performed by a specialist - a mohel. (Until recently, all English royal sons were circumcised by the Royal Mohel. This had nothing to do with religion; the parents wanted the most experienced person to perform the operation and the Royal Mohel was that person.)
So, January 1 of the accepted international solar calendar is, as some Catholics and maybe a few others know, the "Feast of the Circumcision."
I suppose we - Jews - could celebrate a Jewish occasion, but why would we want to celebrate the brit melah of a boy whose followers have caused us so much grief: pogroms, inquisitions, dispositions with foregone conclusions, book burnings, the European holocaust. (That is NOT to suggest that we - Jews - had it much better in Muslim-dominated lands; we did not.)
Jesus, if he really lived, would have been thankful he was born of a Jewish mother rather than Muslim parents; some Muslims, at least in Morocco, still circumcise their sons at age 13 - that's got to be painful at the time and in the boy's memory throughout life.
Now, about "Sylvester"
In Israel many heloni - non-religious Jews - celebrate "Sylvester," an "a/k/a" for New Years' Eve on the philosophy that
כל סיבה למסיבה
or, roughly translated, "any excuse for a party."
The problem with "Sylvester" is that Sylvester was an anti-Semite or, more specifically, anti-Jewish. (Arabs also are Semites.)
According to the English-language Jerusalem Post, Israeli New Years, most commonly known as Sylvester, is named after an anti-Semitic pope.
It's origins, the paper continues, come from Pope Sylvester I, whose saint's day falls on December 31 (the day he died in 335 CE). Pope Sylvester" convinced Constantine to prohibit Jews from living in Jerusalem, and during the Council of Nicea he arranged for the passing of various anti-Semitic legislation.
The holiday "Sylvester, the Post speculates, came to Israel as part of the baggage of European immigrants; Sylvester was celebrated in a number of European, mostly Catholic-dominated, countries.
Meanwhile, ignoring the fact that most Israelis who celebrate "Sylvester" don't know it's origin or who this "Sylvester" was, the rabbis in Jerusalem have banned celebrations on the civil new year's eve.
According to Israel Today, The Chief Rabbinate of Israel has declared that there can be no New Year’s celebration in the Land of Israel. The Rabbinate’s requirements state explicitly that "placing references to Gentile holidays at the end of the secular year is not allowed.” That means no New Year’s Eve for many Israelis.
That ruling, of course, normally would have no impact on the non-religious Jews in Israel who pretty much ignore the rabbis anyway.
The rabbis, in their wisdom - and wanting to punish helonim that plan to celebrate at local hotels and restaurants - ruled that "Any hotel or restaurant offering a New Years Eve celebration will have their Kosher certificate revoked.” Similar warnings were issued to establishments not to display Christmas trees or put up New Year’s decorations.
Israel Today notes that December 31 is a, perhaps "junior" Tisha b'Av.
On January 1 in 1577, Pope Gregory XIII decreed that all Roman Jews, under pain of death, must listen to the compulsory Catholic conversion sermon given in Rome’s synagogues after Friday night services.. The following year (1578), the same pope signed into law a tax forcing Jews to pay for the support of a “House of Conversion” to convert Jews to Christianity and in 1581, Gregory ordered his troops to confiscate all sacred literature from the Roman Jewish community. Thousands of Jews were murdered in the campaign.
The civil calendar the world uses today is known as the "Gregorian" calendar since Gregory XIII revised the earlier Juliann calendar to suit himself. See Does the Pope have the authority to change times and laws?.
For more on "Sylvester" - the pope not the cat - read Do you know what Sylvester is?
