Some people say “Happy Christmas”, but more say “Merry Christmas”. Many people say “Happy new year”, but almost no-one says “Merry new year”. Maybe it is possible to be merry for one day or even 12, but not for 365 (or 366).
The song “We wish you a merry Christmas and a happy new year” dates back to at least the 16th century, which is intriguing, because New Years Day was then generally 25 March. 1 January did not officially become New Years Day in England until 1752. At the beginning of the 19th century “Happy new year” was more common than “Merry Christmas”, but the latter got a boost in 1843, with the publication of Charles Dickens’s “A Christmas Carol”. “Merry Christmas” had surges and falls throughout the 20th century, possibly related to wars and economic conditions, and has doubled in use since the 1970s.
The sometimes-maligned “Happy holidays” has grown in use over the last 30 years, but is still used less than “Happy Christmas” and far less than “Merry Christmas”.
I am not convinced by the use off “merry” in conjunction with “Christmas”. Definitions of and synonyms for “merry” quickly degenerate into the realm of excess drinking. “Happy” and “merry” both appear in the Bible, but not in conjunction with the birth of Jesus. The most appropriate biblical word is probably “blessed”, but “Blessed Christmas” sounds excessive, though it is used slightly more than “Merry New Year”.