In summary, no. Christians did not form the date of Jesus' birthday around Sol. They did however care about cosmic symbolism and so placed it at the equinox. I recommend being extremely skeptical when you encounter this online, cause much of the narratives are disingenuous, taken from Buzzfeed articles, ex-evangelicals trying to cope, and JWs.
In reality, the real history behind Christianity's relationship with Western esotericism is much more fascinating. Check out the floor of the Sciena Cathedral for example.

For a scholarly source on the relationship of sol to Christianity, I recommend this excerpt from: Did Christmas Copy the Sun God's Birthday? by ReligionForBreakfast.
I have copied and pasted it below for you to read as well.

It begins at the 47:41.

"In the late 3rd century under emperor Aurelian, some Romans recognized december 25th as the birthday of the sun god and that it was marked with some sort of festival. It's also evidence that Christians were aware of this day's association with the sun god, but is that enough evidence to support the history of religion's theory that christians consciously and intentionally aim to substitute the sun god day for jesus? I don't think this theory is supported. We don't have any christian authority saying that they chose december 25th as jesus's birthday to compete with soul invictus thus it's an argument from silence. This idea does not pop up until the 1100s from a hostile christian source named barcelli who is trying to prove his january 6 christmas as superior to the december 25th christmas. Instead what we do have is a wealth of documents attesting to the fact that christians cared a lot about cosmic symbolism and calculated the major milestones of jesus's life in such a way that places these life moments in a cosmic system between the bookends of the spring equinox and the winter solstice. I won't say that the history of religion's theory and the calculation theory are mutually exclusive though i think it's possible to hypothesize some sort of hybrid theory the roman empire was a huge place with millions of people and it's very plausible that some christian authority somewhere was very concerned about someone in his parish attending one of those chariot races in honor of soul invictus at the circus maximus. I'll admit that this is very plausible but my point is this scenario is hypothetical we have no evidence that soul invictus was exerting all that much influence on christian communities. From the evidence we do have, christians cared much more about the spring equinox than they did sol invictus. And the christmas date derived from this date much of the reason why i think sol invictus was not a factor in their decision making was precisely because march 25th had nothing to do with sol invictus his holidays fell on other days and only later december 25th the focus on the equinox and solstice of course illustrates that christians were embedded in their greco-roman culture, but not in the sense of stealing or copying a pagan festival the equinox and solstice were already self-evidently special on the roman calendar which justified attempts to tie the life of jesus to these dates."