First of all, it should be clear that Muslims are commanded to deal justly and kindly with their non-Muslim acquaintances, neighbors, friends, and families. Therefore, Muslims are allowed to wish non-Muslims happiness on their festive days. However, they are not allowed to celebrate or participate in the religious festivities of non-Muslims.
For better clarification on this issue, Dr. Mostafa Az-Zarqa said:
When Muslims greet their Christian acquaintances for Christmas, in my opinion, it is by way of compliment and keeping good relations with them. It is not prohibited in Islam to make these acts of compliment and friendliness, especially because Christ is one of the great Messengers of Allah according to our Islamic creed. Thus, he is revered in our religion too, but the Christians exaggerate in revering him and believe that he is Allah, Glorified is He and Exalted above what they say.
Whoever thinks that wishing Christians a merry Christmas is unlawful because it has to do with their belief in Christ’s divinity is wrong, as there is no connection between this compliment and the details of their creed and their excessiveness in it.
Once a funeral procession of a Jew passed by our Prophet Muhammad (peace and blessings be upon him) while he was sitting with his Companions, so he stood up by way of expressing the solemnity of death, and this has nothing to do with the dead person’s belief.
It is required of Muslims to show the merits and moderation of Islam to non-Muslims, and Muslims are not allowed to force non-Muslims to embrace Islam, even if they are living in a Muslim state. Muslims should be tolerant and leave non-Muslims embracing whatever religion they like.
In addition, Muslims today are suffering from a state of weakness among other countries of the world, and major countries plot against them and accuse them of being terrorists and fanatics who cannot be trusted. Therefore, Muslims now need to change this dark image.
Muslims may be visited and greeted by Christian acquaintances in the Muslim feasts, and when the Muslims do not visit the Christians back in Christmas, this will confirm the idea that Muslims are harsh and unwilling to harmonize with others or treat others nicely.
What is said regarding greeting for Christmas applies for New Year’s Day a fortiori, because the latter is not related to belief; it is just the beginning of the calendar.
When `Umar (may Allah be pleased with him) gathered the honorable Companions to decide on an event to be the start point of calculating the Muslim calendar, some proposed following the Roman Calendar or the Jewish Calendar. If that were unlawful, it would not have been proposed.
As long as greeting for Christmas and the New Year’s Day is lawful, printing [non-religious] greeting cards and selling them are lawful too, because the means of the lawful are lawful also.
However, there is a point here that should be noted. Muslims may greet Christians for their feasts by way of compliment and nice behavior, but celebrating the new year in any way that is prohibited is another issue involving following non-Muslims in their traditions and unlawful ways of celebration, which is unlawful.