No, it's almost certainly not because an old Pagan festival was taken over, as popular myth would have it. Biblical Archaeology Review reports:
It’s not until the 12th century that we find the first suggestion that Jesus’ birth celebration was deliberately set at the time of pagan feasts. A marginal note on a manuscript of the writings of the Syriac biblical commentator Dionysius bar-Salibi states that in ancient times the Christmas holiday was actually shifted from January 6 to December 25 so that it fell on the same date as the pagan Sol Invictus holiday. In the 18th and 19th centuries, Bible scholars spurred on by the new study of comparative religions latched on to this idea. They claimed that because the early Christians didn’t know when Jesus was born, they simply assimilated the pagan solstice festival for their own purposes, claiming it as the time of the Messiah’s birth and celebrating it accordingly.
More recent studies have shown that many of the holiday’s modern trappings do reflect pagan customs borrowed much later, as Christianity expanded into northern and western Europe. The Christmas tree, for example, has been linked with late medieval druidic practices. This has only encouraged modern audiences to assume that the date, too, must be pagan.
There are problems with this popular theory, however, as many scholars recognize. Most significantly, the first mention of a date for Christmas (c. 200) and the earliest celebrations that we know about (c. 250–300) come in a period when Christians were not borrowing heavily from pagan traditions of such an obvious character.
Granted, Christian belief and practice were not formed in isolation. Many early elements of Christian worship — including eucharistic meals, meals honoring martyrs and much early Christian funerary art — would have been quite comprehensible to pagan observers. Yet, in the first few centuries C.E., the persecuted Christian minority was greatly concerned with distancing itself from the larger, public pagan religious observances, such as sacrifices, games and holidays. This was still true as late as the violent persecutions of the Christians conducted by the Roman emperor Diocletian between 303 and 312 C.E.
This would change only after Constantine converted to Christianity. From the mid-fourth century on, we do find Christians deliberately adapting and Christianizing pagan festivals. A famous proponent of this practice was Pope Gregory the Great, who, in a letter written in 601 C.E. to a Christian missionary in Britain, recommended that local pagan temples not be destroyed but be converted into churches, and that pagan festivals be celebrated as feasts of Christian martyrs. At this late point, Christmas may well have acquired some pagan trappings. But we don’t have evidence of Christians adopting pagan festivals in the third century, at which point dates for Christmas were established. Thus, it seems unlikely that the date was simply selected to correspond with pagan solar festivals.
The December 25 feast seems to have existed before 312 — before Constantine and his conversion, at least. As we have seen, the Donatist Christians in North Africa seem to have know it from before that time. Furthermore, in the mid- to late fourth century, church leaders in the eastern Empire concerned themselves not with introducing a celebration of Jesus’ birthday, but with the addition of the December date to their traditional celebration on January 6.
There is another way to account for the origins of Christmas on December 25: Strange as it may seem, the key to dating Jesus’ birth may lie in the dating of Jesus’ death at Passover. This view was first suggested to the modern world by French scholar Louis Duchesne in the early 20th century and fully developed by American Thomas Talley in more recent years. But they were certainly not the first to note a connection between the traditional date of Jesus’ death and his birth.
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 died 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.
This idea appears in an anonymous Christian treatise titled On Solstices and Equinoxes, which appears to come from fourth-century North Africa. The treatise states: “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.” Based on this, the treatise dates Jesus’ birth to the winter solstice.
Augustine, too, was familiar with this association. In On the Trinity (c. 399–419) he writes: “For he [Jesus] is believed to have been conceived on the 25th of March, upon which day also he suffered; so the womb of the Virgin, in which he was conceived, where no one of mortals was begotten, corresponds to the new grave in which he was buried, wherein was never man laid, neither before him nor since. But he was born, according to tradition, upon December the 25th.”
In the East, too, the dates of Jesus’ conception and death were linked. But instead of working from the 14th of Nisan in the Hebrew calendar, the easterners used the 14th of the first spring month (Artemisios) in their local Greek calendar - April 6 to us. April 6 is, of course, exactly nine months before January 6 — the eastern date for Christmas. In the East too, we have evidence that April was associated with Jesus’ conception and crucifixion. Bishop Epiphanius of Salamis writes that on April 6, “The lamb was shut up in the spotless womb of the holy virgin, he who took away and takes away in perpetual sacrifice the sins of the world.” Even today, the Armenian Church celebrates the Annunciation in early April (on the 7th, not the 6th) and Christmas on January 6.
Thus, we have Christians in two parts of the world calculating Jesus’ birth on the basis that his death and conception took place on the same day (March 25 or April 6) and coming up with two close but different results (December 25 and January 6).
Connecting Jesus’ conception and death in this way will certainly seem odd to modern readers, but it reflects ancient and medieval understandings of the whole of salvation being bound up together. One of the most poignant expressions of this belief is found in Christian art. In numerous paintings of the angel’s Annunciation to Mary — the moment of Jesus’ conception — the baby Jesus is shown gliding down from heaven on or with a small cross (see photo of detail from Master Bertram’s Annunciation scene); a visual reminder that the conception brings the promise of salvation through Jesus’ death.
The notion that creation and redemption should occur at the same time of year is also reflected in ancient Jewish tradition, recorded in the Talmud. The Babylonian Talmud preserves a dispute between two early-second-century C.E. rabbis who share this view, but disagree on the date. Rabbi Eliezer states: “In Nisan the world was created; in Nisan the Patriarchs were born; on Passover Isaac was born...and in Nisan they [our ancestors] will be redeemed in time to come.” (The other rabbi, Joshua, dates these same events to the following month, Tishri.) Thus, the dates of Christmas and Epiphany may well have resulted from Christian theological reflection on such chronologies: Jesus would have been conceived on the same date he died, and born nine months later.
In the end we are left with a question: How did December 25 become Christmas? We cannot be entirely sure. Elements of the festival that developed from the fourth century until modern times may well derive from pagan traditions. Yet the actual date might really derive more from Judaism — from Jesus’ death at Passover, and from the rabbinic notion that great things might be expected, again and again, at the same time of the year — than from paganism. Then again, in this notion of cycles and the return of God’s redemption, we may perhaps also be touching upon something that the pagan Romans who celebrated Sol Invictus, and many other peoples since, would have understood and claimed for their own too.
There's more at the link.