No, that's not right. Christmas is December 25th on both calendars, but December 25th on the Julian calendar is (currently) January 7 on the Gregorian calendar. The Gregorian calendar reform didn't "set the nativity at December 25," but it removed 10 days (at the time) from the calendar to put the calendar's relationship to the seasons back where it was at the time of the Council of Nicaea.
Since then, the two calendars have continued to diverge. The difference is now up to 13 days, and will go to 14 days in 2100.
It depends on your frame of reference.
Most people and societies use the Gregorian. Orthodox except for Greek Orthodox celebrate the Nativity on January 7 under the Gregorian.