No. Say whatever is in your heart to say, hopefully with some consideration for your audience. Likewise, if you hear a heartfelt greeting other than your preferred one, it's up to you to smile & not be offended, unless the otrher person is deliberaely trying to goad you; then I'd hold back on a reaction just to be a better person. I'm Jewish & I'll say "Merry Christmas" to those I know will appreciate it.
If someone says "Merry Christmas" to me, I hear, "I wish for you the same joy in your holidays as I wish for my own." & I can, from the heart & honestly, respond, "Thank you; you, too!"
If they're a prig trying to start something, I *MAY* respond with an annoying lecrture on how there really are "holidays," even for Christians (1st day od winter, Xmas Eve, Xmas, Boxing Day, New Year's) so, pedantically speaking, "happy holidays" is appropriate.
When I volunteer for the Salvation Army's Kettle Campaign, I say, "Merry Xmas," to all, because I'm representing the Sallies & that's their focus. . .