Christmas hasn't been a religious holiday in the US for decades. Christmas is a commercial holiday where from disparate religions and all parts of the country get together in a month-long worship of O' Mighty Dolla. The worship is most focused in specially-designed temples to O' Mighty Dolla, called 'shopping malls'.
To somebody I haven't met, I usually smile and say whatever greeting first comes to mind, be it "Happy Holidays", "Merry Christmas", "Seasons' Greetings", or just "Hello." From my perspective, the terms are all basically equivalent, and I treat them interchangably.
To a friend, I'll take the time to learn their preference, and generally use that as a sign of respect, provided they have a valid reason to prefer one over the other.
To somebody who gets annoyed at the use of a term, and feels entitled to do so at my expense, I'll use the term at every opportunity. It is not my fault that you have chosen to become offended at something, nor is it my responibility to satisfy your sense of entitlement by adhering to your world-view. If you choose to become offended every time somebody says "Merry Christmas" to you, then don't be surprised if I say "Merry Christmas" every time I see you, in order to point out the absurdity of your irritation.
If, instead, when a stranger smiles at you and says "Merry Christmas", you have the ability, whether you believe in the divinity of Christ or not, to simply smile and return what was meant as a warm-hearted greeting, you will make it through the commercial holiday season a much happier person.