Well … Holiday comes from the Old English hāligdæg, meaning holy day. So it closer to it than one might think.
But on the other hand … conservative christians might not get that connection
This usually gets downvoted but I love it when people remind Christians that all of their holidays are bastardized versions of other cultures holidays.
It's when you're dealing in an official capacity or speaking to a broad audience or when you don't know the person's culture.
The CEO saying Merry Christmas to his 140 employees, when 5 of them are Jewish is going to be not feel so great for those 5 Jewish people. Happy Holidays should be fine for everyone.
But if you know the person is Christian (or celebrates Christmas) it should be perfectly fine to tell them Merry Christmas.
Of course in some places that may be considered insensitive because a Jewish person might be hearing it. Which is absurd and that level of sensitivity is not acceptable IMO.