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, because both of these matters involve imitation of the kuffaar, 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 Dawood, 4031; classed as saheeh by al-Albaani in Saheeh Sunan Abi Dawood
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.