Why is Christmas Day on 25 December?

The celebration of the birth of Christ on 25 December dates back to the 4th century, when Theodosius I made Christianity the official religion of the Roman Empire.

Constantine (who was only baptized on his death bed) lived in the 3rd and 4th century, ruling roughly from 306 to 337 AD. It was emperor Theodosius I ruling in the late 4th century who made Christianity the de-facto official religion of the Roman Empire. Nor was Diocletian the last pagan emperor. Constantine the Great (who came after Diocletian) was in all probability initially a believer in Sol Invictus and was only influenced towards Christianity by his mother Helena (who was Christian). The last pagan emperor was Julian, known as the apostate or (depending on your point of view) the philosopher. He ruled for a few years in the middle of the 4th century and was the last scion of the house of Constantine, being the youngest son of Constantine the Great's half brother.

Special thanks to Thomas Fiedler

Also

"Christianity usurped all the [pagan] holidays and festivals, attributing some Christian marvel to each solstice festival, attaching new names to [some] such as the Eve of the Dead, Day of the Dead, Morrow of the Dead (Halloween, All Saint's, All Souls). [A]nd it is mightily uncanny that the [Christmas] holiday just happened to hit [at] the same time as the [winter solstice. A date recognized for] many thousands of years [...] long before Christianity even existed. "

[...]

"The date was chosen to coincide with the winter solstice and the Roman festivals associated with the shortest day of the year, which falls between 22 December and 25 December. This was seen as the day when the Romans celebrated Dies Natalis Solis Invicti – "the birthday of the unconquered sun". It was also Jupiter's birthday and, further back, the birthday of his Greek equivalent, Zeus."

Special thanks to Kay