“Xmas” has been used in English for centuries. The X is an abbreviation for Christ, from the first letter of Greek Christos. First appearing in English in the twelfth century, it was spelled with Xp or Xr, corresponding to the Greek “Chr,” thus Xres mæsse meant “Christmas.”
Haliday, a thirteenth century term, came from Old English haligdæg, or holy day from halig “holy” and dæg “day.” The meaning “religious festival” came to be equated with something like “day of recreation.” In American English, “the holidays” are exclusively associated with Christmas and the new year. The religious roots are undeniable, so “happy holidays” is as authentic a Christmas greeting as any other.