Read well, you who would baselessly assert that Christmas is pagan.
Ok... let's go over this again. There is no evidence Christmas is pagan.
I'll start with the date.
Sol Invictus was not placed on 12/25 until 354 AD when the Philocalian Calendar records this but doesn't specify any festival with regards to sun worship. Prior to this, the Julio-Claudian fasti inscriptions say Sun festivals were on August 8th, 9th, 28th, and December 11th, and maybe October 19th. The Philocalian Calendar says Emperor Aurelian honored the sun with chariot races every 4 years Oct 19-22 (Steven Hijmans, "Sol Invictus, the Winter Solstice, and the Origins of Christmas", Mouseion, Number 47/3 (2003), 277-298).
Saturnalia was never on 12/25. Macrobius says Saturnalia began 14 days before January, which comes out to December 17th (Saturnalia 1.10.1-23), using Roman Calendrical dates. He says it lasted for 3 days, but according to the Fasti inscriptions, it lasted to the 24th during the days of the Republic.
Yule was placed on 12/25 by King Haakon the Good in the 10th century AD to coincide with Christmas. This goes back to the early historian Snorri Sturluson, and his book "Heimskringla: History of the Kings of Norway." Snorri says before this, “Yule was celebrated on a midwinter night, and for the duration of three nights" (p. 106). He gives no specific dates, but St. Bede in “The reckoning of Time,” (Section 329) said Northmen calculated their seasons according to the cycles of the moon, so the date of Yule probably changed every year. Pliny the elder also says the Gallic tribes calculated their months according to the moon (nat. his. 16.95.250). Last, according to the Chronicler, Theitmar, the Danes sacrificed to pagan gods in January after the 6th (The Chronicon of Thietmar of Merseburg, Book 1.17).
The Winter Solstice had no festive significance to ancient Romans. There were no celebrations planned for the date and they disagreed on when it was. The Julian Calendar does say 12/25, but Pliny the Elder says 12/26 (nat. his. 18.59.221), and Columella says 12/23 (De Re Rustica 9.14.12).
The date of 12/25 was selected by Christians because (according to early authors like Dionysus Exiguus & Hippolytus) they believed Jesus was conceived on the same day he died and they thought he died on March 25th, so just count forward nine months.
Christmas trees only go back to the 16th century in eastern Europe. The first mention of Christmas trees is in an Alsace ordinance in 1561. Almost no early pagans thought pine trees were sacred, let alone associated with 12/25. Germanic tribes believed the oak was sacred (Pliny, nat. his. 16.95). Maximus of Tyre said, "The Celts indeed worship Zeus, but they honor Him in the form of a lofty oak" (Dissertation VIII, section 8).
The most likely explanation is they morphed over from paradise trees. Adam and Eve's festival days is 12/24 and used to be honored with an Adam and Eve play. In Europe, in the dead of winter, not a lot of trees are available, so they would get a pine tree and decorate it with fruits and cakes. After the play, they could eat the treats on the tree and the practice probably just morphed into Christmas. We have no record of Christmas trees prior to this time. See "Christmas: A Candid History" by Bruce David Forbes, pp. 48-59; Encyclopedia of Christmas by Tanya Gulevich, pp. 165-171.
Santa Claus can only be traced back to Dutch immigrants in New York in the early 1800s. He came from the Dutch Sinterklaas, also known as St. Nicolas. His feast day was on 12/6 and was moved to 12/25 around this time to help make Christmas a family holiday. Newspapers promoted it and encouraged to give gifts on Christmas instead of New Years' like it traditionally was before this. Sinterklaas was rebranded from a catholic priest to look like a traditional dutchman from that time period, which included a big red suit (A dictionary of English Folklore, section: Santa). After this Santa was exported around the world and different countries added new spins. Stockings were also promoted around this time period and trace back to Clement C. Moor's “A Visit from St. Nicholas.” In 1927, in Finland, a radio broadcaster, Markus Rautio, morphed the old pagan deity, Joulupukki, into a Santa figure. Santa changed the Joulupukki figure, not the other way around.
Father Christmas, surprisingly, pre-dates Santa Claus. But he is not a pagan deity. Instead, he was a medieval personification of Christmas. Richard Smart of Plymtree is the first to write about him, referring to him as Sir Christmas, and his task is to announce the birth of Christ (Oxford Book of Carols, no. 21, 41-3).
Mistletoes do not go back to paganism. The first we hear of using mistletoe as Christmas decorations are in Robert Herrick, "Hesperides poetry collection," (892, 980), and then William Coles mentions it is a decoration in “The Art of Simpling” (1600s). The tradition of kissing under it did not begin until the end of the 18th century in Britain. See "A dictionary of English Folklore," section: Mistletoe.
Yule Logs do not go back to paganism, despite the name. Yule is also an English word to mean "mid-winter period." The first mention of yule logs is in Robert Herrick, "Hesperides poetry collection (number 784)" and he calls it a Christmas log. It wasn't called a yule log until Aubrey's “In the west-riding of Yorkshire on Christmas eve” (1686, p. 134).
St. Francis is credited for the first nativity scene in 1223 AD, Greccio. His followers then promoted the use of them during Christmas (Life of Saint Francis of Assisi - Saint Bonaventure. Chapter 10.7).
There is no evidence anything associated with Christmas goes past these sources I mentioned here, and therefore, there is no evidence Christmas is pagan.