William J. Tighe in his article, Calculating Christmas (reprinted at my church's website), the reverse is true.
The Roman Emperor Aurelian instituted on the 25th of December in 274 a festival known as the Sol Invictus or "Birth of the Unconquered Sun" for all Romans. The emperor did this to coincide with the Winter Solstice, when the days begin to grow longer, as an analogy of the rebirth that he hoped to see in Rome which was collapsing in on itself.
A Roman Studies specialist, Dr Steven Hijmans, in his article, Sol Invictus, the Winter Solstice and the Origins of Christmas, goes a bit further claiming, "the suggestion that it was established by Aurelian cannot be proven. In fact, there is no firm evidence that this feast of Sol on December 25 antedates the feast of Christmas at all." He also claims that many of the feast days for Sol were being ignored by 354.
Besides the Sol Invictus celebrations, there were only two temples in Rome dedicated to the sun. One of the temples celebrated it's festival on August 9th, which was maintained by Aurelian's clan, and the other celebrated it's festival on August 28th. Neither temples had festivals associated with solstices or equinoxes.
If Sol Invictus is not the source of the Christmas date, how did Christians choose the 25th of December?
Two of the Gospel books tells of Christ's birth but provide no date. In all of the New Testament there is no date provided for Christmas. Later apocryphal writings will add their own details the the birth narrative but also do not provide a date. Around 200 AD Clement will make mention of various churches that attribute different dates for His birth and not one of them is the 25th of December. So a lot of interest, but no certainty. However, by the 4th century two dates gained prominence. December 25th by the church in the west and January 6th by the church in the east.
Why those two dates?
There was a widespread Rabbinic concept in Judaism known as the "integral age" of prophets that they died on the same day they were either born or conceived. It is the day of Christ's death that becomes the key factor of when He was born.
Early Christians knew the date of Christ's death to be the 14th of Nisan, in accordance with the Jewish Lunar calendar. But as Christianity spread out of Jerusalem it encountered other cultures with various forms of calenders, some based on the moon, some based on the sun. The Greeks correlated 14 of Nisan with their calendar, making it the 14th of Artemision. This calendar would be superseded by the Roman calendar, changing the date to April 6th. Christians in the west would go through their own calendar issues and end up with the date March 25th.
Add the typical length of pregnancy, 9 months, and you get December 25th and January 6th. Saint Augustine was familiar with this dating method as revealed in his, On the Trinity, where 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."
The western date of the 25th would be adopted by the east as well as the date of Christmas. The eastern date of January 6th would be fixed specifically to celebrate Epiphany, which would be adopted by the east. There are 12 days between Christmas on the 25th and Epiphany on January 6th: The Twelve Days of Christmas.
While it is highly unlikely that Jesus was born on December 25th, the two dates believed to be the dates of Christ's death, March 25th and April 6th, coupled with the Rabbinic concept of the intergral age is how early Christians decided on the date of Christmas, not the Sol Invictus festival.
* This subject is covered in more detail by Andrew McGowan, in his article, How December 25 Became Christmas.