Strangers in the Night - S6-E12
Character mistake: The whole idea of the serenading scene was Larry was supposed to hide and sing for Jack while Jack would be shown in the yard alone and pretend to be the one singing, but the yard was brightly lit and Larry was right behind Jack and both were very shown with nothing to hide Larry.
Answer: He's not a Catholic priest. He's a pastor, with the title Reverend, at a community church who chooses to wear a clerical collar. Some people still address pastors as "Father" though. I'm not sure of his religious denomination, but most pastors can marry.
Bishop73