I am watching the re-play. Giuliani was just bragging about all the weddings he did as Mayor of New York. Even though he was a public official - as a Catholic, is he supposed to be performing weddings?
Civil ceremony ok, religious ceremony no.
keep in mind in most states the Priest IS ALSO a notary so the priest can notarize the marriage license. It is a matter of convenience.
It is just a solenization before a notary that counts for the public recording.
This way if the marriage produces children, we have accountings for inheritance and taxes.