Not that it makes the school’s decision any more acceptable, but Davidson College is a private university, not public.
But does it get federal dollars?
Actually, that makes the school's decision entirely more acceptable. Chick-fil-A has the right to express its opinion, and customers have the right to vote with their wallets and decide whether or not to eat there. Davidson College, as a private university, has the right to (a) express its opinion about Chick-fil-A and (b) decide whether to spend its money at Chick-fil-A; likewise, students and prospective students have the right to vote with their wallets and decide whether or not to attend school there.