Ray Bradbury is associated more with autumn and Halloween than he is with Christmas, but he did write a couple of stories with a Christmas theme. Well, sort of...
"The Man", first published in Thrilling Wonder Stories in February 1949 (and later collected in The Illustrated Man, S is for Space and Bradbury Stories) tells of a space crew which arrives on an alien planet expecting to be celebrated or greeted. But the local inhabitants have no interest in them. They've arrived with very poor timing, as the aliens are in awe of some much more interesting guy who arrived yesterday. Not just any guy, but "a remarkable man, [...] good, intelligent, compassionate, and infinitely wise!"
A man for whom they had waited a very long time...
Once convinced of the special capabilities of "The Man" (healing is involved...), the captain vows to follow him from planet to planet, hoping to one day catch up with him. The story never actually says who "The Man" is, but you're clearly supposed to see him as Jesus (or some deity of your choosing).
Curiously, a couple of years later in another pulp magazine, a lesser writer tackled the same idea. But in a most literal and obvious way. Charles E. Fritch wrote "Night Talk", published in Startling Stories in September 1952. In this story, a rocket ship makes a bumpy landing on Mars, and the pilot makes his way to the nearest hotel - like you do - and tries to get a room for the night. The hotelier tells of how he once made a mistake in turning away a couple from Earth, telling them there was no room at this particular inn.
Fritch's story is quite forgettable, but thankfully brief. If you're interested it, you can find it at the Internet Archive, here. Amusingly, SF writer and critic James Blish at first believed this to be a Bradbury story published under a pseudonym. Mars? Check. Primitive but metaphorical description of a rocketship? Check. Earth destroyed so everyone's trying to get to Mars? Check. Re-use of the idea of a messiah travelling from planet to planet? Check. All the clues were there. But Blish was wrong.
Blish can be forgiven. Fritch had begun publishing only in 1951, and his Bradbury-influenced Mars story could easily have been a lesser Bradbury; and Bradbury had published under pseudonyms. However, Fritch turned out to have a long, if not particularly illustrious career, publishing his last works in the 1990s. Blish's error - and correction of the error - can be found in his collection of SF reviews, The Issue at Hand (1964, under Blish's critic pseudonym William Atheling Jr.)
Blish's reason for discussing the two stories - the Bradbury and the Fritch - is to make a point about the importance of characters in good fiction. Fritch's central character, just called "the traveller", has no distinguishing characteristics. Bradbury's central character, Captain Hart, is just your average 1940s pulp magazine space hero. Blish suggests that both Fritch and Bradbury could have learned a thing or two from Anatole France's story "The Procurator of Judea", which brings to life one Aelius Lamia and a certain Pontius Pilate. France's story has no plot connection to "The Man" or "Night Talk", but like both of those stories it places the Christian Jesus into the background of a story for the purposes of irony or awe, depending on your religious persuasions.
You can read "The Procurator of Judea" here.