[Rhadaghast]

Yes, there is documentation of a pagan teaching changing the tradition of the RCC.

On what day was Jesus born?

The Jewish Christians can determine this definitively to be in our month of March.

Who was born on Decemeber 25th?
Before the Progression of the equinox, 12-25 was the birthday of Tammuz. Ezekiel 8 documents this practice as being in the temple.
Nimrod's Son Tammuz born to the Queen of Heaven. This was a world wide cult of Sun Worship.
When Tammuz died he was remembered by 40 days of his followers denying themselves a pleasure from this life, culminating in the celebration of Tammuz mother's return to Earth as the Goddess of fertility on Easter.
When she demonstrated her divinity by changing a bird in to an egg laying rabbit.