Is it permissible for Muslims to exchange greetings and prayers with one another on the occasion of the New Gregorian Year without the intention of celebrating it?

It is not permissible for Muslims to exchange greetings or celebrate the Gregorian New Year, as this involves imitating the disbelievers. The Prophet, peace and blessings be upon him, said: "Whoever imitates a people is one of them." Furthermore, congratulating someone on a day that recurs annually falls under the meaning of celebrating it and taking it as a holiday, which is forbidden.