The claim Jesus was born on another day doesn't change anything on the significance of Christmas. The Church celebrates Jesus’ birth on December 25, but this doesn’t amount to a claim he was born on that day.

The liturgical commemoration of an event doesn’t mean the Church holds it happened on that day. For example, the day a saint is commemorated is frequently the day of his death, but not always. 

One will find Church documents referring to the liturgical celebration of Jesus’ birth on December 25, but one won’t find any magisterial documents establishing it as a Church teaching that this is when he was born.

Though his birth has profound significance for our faith, the particular day it occurred is a matter of history rather than doctrine, and Christians needn’t be disturbed by the idea he was born another day.