OTL, where Christian were in important numbers in Islamic countries, Muslims took the use of celebrating with them (not between Muslim only) Christmas.
While it was heavily criticised and condamned by religious head or islamic scholars, it benefited from the coranic prescription "If your neighbour is celebrating something, celebrate with him".
Now, having a TL where Christmas become a Muslim holy day...
It's hard, but not totally impossible. You'll probably need to butterfly the most rigorist school of Islam or at least prevent them to reach the most "christianised" Islamic regions (Al-Andalus, Palestine, Syria, Anatolia...)
After some time, you might have some schools (the same allowing consumption of alchool) saying "oh, WTH, let's do it". Still, it would be REALLY hard to have it even in the better conditions except if you manage to get a religious unity based on a national feeling (and so, require an Islamic country with a christian majority (at least plural) to survive up to XVII century.