In which century did December become the month in which Christmas is celebrated?

Answer 1

The most obvious answer is that the idea, purpose and date for Christmas (December 25th) was to provide continuity with the winter solstice traditions of many cultures. By doing this converts could continue to have a traditional midwinter celebration even if they had to celebrate in the "new" Christian church. Many scholars indicate that an actual Christmas Day celebration would be between May and July.

Interestingly the celebration of birthdays -- even including that of Christ -- was rejected as a pagan tradition by most Christians during the first three hundred years of Christianity. In reaction to the claims by Gnostics that Jesus had not been mortal, and to "humanize" Christ Christians began to emphasize the Nativity. As this reworking continued many third century Christians supported observance of Christ's birthday, but condemned customs such as exchanging of gifts and decorating homes with evergreens citing Book of Jeremiah:10 which condemns the heathen practice of cutting a tree from the forest to "deck it with silver and gold".

Now, to answer your question:

The December 25 date may have been selected by the church in Rome in the early 4th century. At this time, a church calendar was created and other holidays were also placed on solar dates

Answer 2

In the fourth century C.E., despite the stand taken by the early Christians against the custom of celebrating birthdays, the Catholic Church instituted Christmas. The church wanted to strengthen its position by removing one of the main obstacles in its way​—the popularity of the pagan Roman religions and their winter solstice festivals. Each year, from December 17 through January 1, “most Romans feasted, gamed, reveled, paraded, and joined in other festivities as they paid homage to their deities,” says Christmas in America, by Penne L. Restad. And on December 25, the Romans celebrated the birth of the Invincible Sun. Instituting Christmas on that day, the church cajoled many Romans into celebrating the birth of Jesus instead of the birth of the sun. Romans “were still able to enjoy the trappings of these midwinter festivals,” says Santa Claus, a Biography, by Gerry Bowler. In reality, they “continued to mark the new days with old ways.”