The December 25th birthday is not a date which is given in the Gospels, but rather it is a traditional date that is assigned to the birth of Jesus, as related to the prior pagan traditions. Dec. 25th is in fact a general date that honors the Winter Solstice period (generally from Dec. 22nd to Jan. 6th, depending on the precession of the equinoxes) which is mythologically the traditional death/rebirth of the Sun.
As Frazer denotes:

"In the Julian calendar the twenty-fifth of December was reckoned the winter solstice, and it was regarded as the Nativity of the Sun, because the day begins to lengthen and the power of the sun to increase from that turning-point of the year. The ritual of the nativity, as it appears to have been celebrated in Syria and Egypt, was remarkable. The celebrants retired into certain inner shrines, from which at midnight they issued with a loud cry, “The Virgin has brought forth! The light is waxing!” The Egyptians even represented the newborn sun by the image of an infant which on his birthday, the winter solstice, they brought forth and exhibited to his worshippers. No doubt the Virgin who thus conceived and bore a son on the twenty-fifth of December was the great Oriental goddess whom the Semites called the Heavenly Virgin or simply the Heavenly Goddess; in Semitic lands she was a form of Astarte... Thus it appears that the Christian Church chose to celebrate the birthday of its Founder on the twenty-fifth of December in order to transfer the devotion of the heathen from the Sun to him who was called the Sun of Righteousness.(Jesus)"

[ Frazer, James.: The Golden Bough, Touchstone, 1963. Page 415-417 ]

Now, with this point understood, it is argued that the traditional notion of Jesus being born on Dec. 25th is not a true attribute of Jesus, as the Bible itself doesn't exactly say when he was born; therefor he is not mythologically related to any other sun-god who shares the winter solstice birthday.

This is true in and of itself, but when one steps back to look at the mythological tradition, coupled with the symbolism of the events that surround the birth sequence in the Gospels: such as the Star in the East; Bethlehem(House of Bread); the Three Wise Men(or Magi or Kings); the fact that in the Gospel of James Jesus' birth occurred in a cave* which is also the birth place of Mithras on Dec. 25 (and many other gods prior); along with the 3 day death and resurrection of Christ, (which mythologically reflects the dual birthdays as seen in the Osirus/Horus myths); along with, as Frazer points out above, the "nativity" notion with the "Virgin bringing forth the new born child.." which coincided with the birth of the sungod at the winter solstice, ect.....
With these attributes in consideration, the suggestion that the Dec. 25th birthday is a mere interpolation and nothing more, loses its tangibility. The fact is more likely that the inner circles of the high societies who understood the true myth of Christ, assigned the Dec 25th birth date not just to compel other pagans to join in the trend, but because the very nature of the Christ myth dictated it.

* The cave is symbol for the Sun in the womb of mother earth at the Winter Solstice before it is going to be born-again. The cave Jesus was placed in after his crucifixion before being resurrected or " born again" is the same symbolism.

As G. Massey elaborates:

[ Massey, Gerald.: The Historical Jesus and the Mythical Christ, Book Tree, Pages 40-43 ]