I just don't see how it demeans Christmas to say "Happy Holidays." Christmas is a subset of Holidays in my mind. You're not ignoring it, you're just speaking more generally, that's all. I think those Fox News types are just control freaks who have a burning need for their own personal religious beliefs to be acknowledged by others in an overt way. If it's not, they take great offense. Which is thoroughly childish. Who gives a f&%ck? It's their problem, not mine.
What would I say to someone who is of some other religion than Christian on Christmas day? I wouldn't acknowledge the day at all. Just behave like it's any other day. So my greeting would be more like, "Hey, good to see you." Or "Hey, howse is going?" That's probably what I would say to anyone who isn't family, actually.
Here's a question I have for folks: I was talking to a friend of mine, who happens to be Jewish, a few years ago during the holidays. She was complaining about all the Christmas shopping she had to do. I was shocked! I said, "What the hell are you Christmas shopping for? You're not obligated in the slightest to honor this holiday. You've got a free pass to avoid it entirely." I think she thought I was rude when I was trying to be funny, but I felt like I had a point, too.