[Cronos]

odawg is 100% correct - “[Christmas Day] was the old pagan holiday celebrating the winter solstice and the birth of the sun god and celebrated when the days began to get longer.”
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Modern scholarship has debunked that. Christians, rightly or wrongly as far as the day, were celebrating December 25 before the Romans used the day to celebrate their paganism.

1. People say it was Saturnalia, but that's false
   2. Saturnalia was not celebrated on December 25th; it was, in fact, originally celebrated on December 17th and was later expanded to six days of celebration spanning from December 17th to December 23rd.
      
      By the time December 25th rolled around, Saturnalia was already over. No one in ancient times ever thought Saturnalia was on December 25th.
      
      Nevertheless, the claim that Saturnalia was celebrated on December 25th has somehow become widely propagated online, even though it is just flat-out wrong.
      
      
   3. Saturnalia was not a celebration of the winter solstice. In fact, at least on its original date, it did not even occur on the day of the solstice or even on a day that was believed by ancient writers to have been the solstice. The Julian calendar was poorly calibrated, so the exact date of the solstice changed over the course of many years
   
   
   
2. Next, Sol invictus
   
   1. There is only one surviving ancient source that ever mentions the birth of Sol Invictus being celebrated on December 25th, the Chronograph of 354, an ancient Roman calendar for the year 354 AD. Now, it just so happens that the Chronograph of 354 was written by a Christian - Furius Dionysius Filocalus. and it is also the first source to mention Christmas being celebrated on December 25th.
   2. The cult of sol invictus post-dates the wider celebration of December 25th - January 6th. In fact the cult of Sol invictus really became big only during the troubles of the 3rd century and was nearly state religion under Aurelian

So your "goodyness of Gad" webite, sontagged, has got this one glaring error. Let's dissasemble the rest of the argument

[Varda]

The Bible says Jesus birth was in winter. Shepherds were in the field around Bethlehem. That’s winter (rainy season). December would be the start of lambing season.

Also primitive people nearly everywhere have been able to pinpoint the soltices, some at least since the Neolithic (ex: Stonehenge). It makes no sense that a society as advanced as the Romans would be unable to do so.