According to Charlemagne's biographer Einhard, he didn't choose it at all-- he went to Rome to help the pope out because the people in Rome gauged his eyes out, and then Charlemagne just went to church on Christmas day and was crowned emperor.

We don't have a lot of evidence besides Einhard's account, but it's not very likely that it was because it was a public holiday. They didn't have workdays the same way we do, and Christmas wasn't as much more important than other feast days to medieval people as it is to most Christians now.