New Year’s is a Christian holiday.
The Rama in Darchei Moshe Y.D. s. 148 in name of Terumat Hadeshen holds that even in our times, if a Jew wishes to give a present on the eighth day after Christmas which is called New Years, he should send the present during the day prior and not on the day of the holiday itself. And if the day before the holiday falls out on Shabbat, one may send on the day of the holiday, itself as there is a matter of eiva if one sends later than that or more before then.
Reb Moshe in Iggerot Moshe Even Haezer 2:13 holds that although it is mutar m’ikar hadin, baalei nefesh should be machmir about celebrating this day.
We, the citizens of the world of Torah and kedusha certainly must take the initiative of being baalei nefesh. After all, those who are not baalei nefesh are nebech soulless creatures, literally and figuratively.
More so, to steal our time honored greeting of Rosh Hashana, and use that expression of “Shana Tova” – is insulting, since you are placing a tumahday on equal footing as a holiday.
Boruch mamvil is said even bain kodesh l’chol, kal v’chomer, bein kedusha l’tumah.