All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
We have already clarified in Fatwa 82006 that it is not permissible to congratulate the Christians, Jews or other polytheists on their celebrations as this is confirming the falsehood which they are on.
As regards your statement that it is rude not to congratulate them, then the answer is that if a Muslim behaves in a good manner with them throughout his stay with them, they would certainly have known that he is not rude and he is not a person of a bad character even if he does not congratulate them on the day of their celebration. Indeed, being rude is when one disobeys Allaah. It is an obligation on a Muslim to fear Allaah and endeavour in calling the non-Muslims to Islam but not be kind to them at the expense of his religion and creed.
It should also be noted that pleasing Allaah comes in priority over pleasing His creation, and that whoever pleases people by ensuing the wrath of Allaah, Allaah will be angry with him and He will make people be angry with the person.
Finally, for more benefit on the relationship of a Muslim with a non-Muslim, please refer to Fatwa 88293.
Allaah Knows best.