Praise be to Allah.
It is not permissible to celebrate Christmas or to participate in it in any way, because it is an innovated and fabricated festival, and doing so is implicitly approving of the disbelief of the Christians when they said that the Messiah is the son of God who came down to redeem humanity from sin, and that he was killed and crucified.
This is blatant disbelief and rejection of the Book of Allah, may He be exalted. Hence celebrating this festival is a grave evil.
Adh-Dhahabi (may Allah have mercy on him) said: If the Christians have a festival, and the Jews have a festival, that is exclusively theirs, then the Muslim should not participate in it with them, just as he does not follow the teachings of their religion or face the same direction as them when praying.
End quote from Tashbeeh al-Khasees bi Ahl al-Khamees. Published in Majallat al-Hikmah, issue no. 4, p. 193.
See also the answer to question no. 145893.
There is nothing wrong with talking about ‘Eesaa (Jesus – peace be upon him) at the time when the Christians are celebrating his birth, without participating in any of their celebrations, such as if one speaks on social media, for example, or in classes or in khutbahs from the minbar, explaining that he is a Prophet and Messenger, one of the Messengers of strong will; that he is not God or son of God, and that making such a claim constitutes disbelief in Allah, may He be exalted; and that no one can attain salvation except by following our Prophet Muhammad (blessings and peace of Allah be upon him) and entering the religion of Islam.
Such words are a disavowal of shirk and its people, and denouncing the grave evil that happens at this time. So there is nothing wrong with saying such things; in fact, this is an act of worship and obedience that is prescribed, and it comes under the heading of enjoining what is right and forbidding what is wrong.
And Allah knows best.