Praise be to Allah.
It is not permissible for the Muslims to exchange greetings on the occasion of the Gregorian New Year , and it is not permissible for them to celebrate it. Both of these matters involve imitation of the non-Muslims, and we have been forbidden to do that.
The Prophet (blessings and peace of Allah be upon him) said: “Whoever imitates a people is one of them.” (Narrated by Abu Dawud, 4031; classed as sahih by al-Albani in Sahih Sunan Abi Dawud)
Moreover, offering greetings on this day that comes back each year comes under the heading of celebrating it and taking it as a festival, which is also forbidden.
And Allah knows best.