If you frequent social media, then you’ve probably been inundated with smarty pants pseudo scholars calling Christmas a “pagan holiday.” These under-informed accusers believe they have the goods with bold claims like Christmas trees are forbidden in Jeremiah 10 (a passage about the manufacture of Canaanite idols) and that the date of December 25th was adopted from the pagan holiday of Saturnalia (actually December 17th).
The best way to respond to these claims is to ask them what evidence they have (from source documents). Most of the time this question will cause them to change the subject or elicit the chirping of crickets. The truth is most of the claims are dubious and have no actual evidence. Even if there were other holidays on the same date, it does not mean that they were even aware of each other. Correlation does not necessarily imply causation. The leaps in logic are prohibitive and there is no source claiming the Christians appropriated pagan days until the 12th century!
In the real world. reputable ancient source documents lay waste to the popular pagan origins theory concerning December 25. Biblical Archaeology Review is a serious journal respected by real biblical scholars. The article How December 25 Became Christmas provides ample evidence debunking the pagan origins myth, showing how it actually began, and even demonstrating how 12-5 was decided to be the date of Jesus’s birth. This dates back to the second century ( I challenge pagan theorists to antedate it).
Around 200 C.E. Tertullian of Carthage reported the calculation that the 14th of Nisan (the day of the crucifixion according to the Gospel of John) in the year Jesus diedc was equivalent to March 25 in the Roman (solar) calendar. March 25 is, of course, nine months before December 25; it was later recognized as the Feast of the Annunciation—the commemoration of Jesus’ conception. Thus, Jesus was believed to have been conceived and crucified on the same day of the year. Exactly nine months later, Jesus was born, on December 25.
Of course, I am a nerd with Logos Bible Software and a searchable database of ancient documents including Tertullian. I verified the claim from the original source documents, Tertullian wrote:
And the suffering of this “extermination” was perfected within the times of the lxx hebdomads, under Tiberius Cæsar, in the consulate of Rubellius Geminus and Fufius Geminus, in the month of March, at the times of the passover, on the eighth day before the calends of April (March 25)
The belief was that he died on the same day he was conceived. They believed the conception was on March 25 which became the Feast of the Annunciation, commemorating the angel’s appearance to Mary: “In the sixth month the angel Gabriel was sent from God to a city of Galilee named Nazareth” (Luke 1:26) This idea appears in an anonymous Christian treatise titled On Solstices and Equinoxes from fourth-century North Africa:
“Therefore our Lord was conceived on the eighth of the kalends of April in the month of March [March 25], which is the day of the passion of the Lord and of his conception. For on that day he was conceived on the same he suffered.” 
The rationale was innocent enough, just add nine months to date of conception to determine when He was born and it lands on December 25th. It’s that simple. Paganism had absolutely nothing to do with it.
Note: Due to so may folks missing the point, I do not think Jesus was born on Dec 25th; the point of this post is that the early church did not chose the date based on a pagan holiday but rather when they assumed he was conceived (mistaken or not).
Andrew McGowan, “How December 25th Became Christmas ” Biblical Archeology Review, 08/12/2014 http://www.biblicalarchaeology.org/daily/biblical-topics/new-testament/how-december-25-became-christmas/#location1
 Tertullian, “An Answer to the Jews,” in Latin Christianity: Its Founder, Tertullian, ed. Alexander Roberts, James Donaldson, and A. Cleveland Coxe, trans. S. Thelwall, vol. 3, The Ante-Nicene Fathers (Buffalo, NY: Christian Literature Company, 1885), 160.
 De solstitia et aequinoctia conceptionis et nativitatis domini nostri iesu christi et iohannis baptista as cited by