I have two favorite Father Mulcahy moments.
The first is in "Dear Sis." A number of wounded soldiers come in during the night, and one of them wants a doctor and refuses to be seen by anyone else. When Mulcahy tries to calm him down, the soldier punches him. Mulcahy, already irritated by resent events, ends up punching him so hard he rolls off his stretcher.
The second is in "Captains Outrageous." Father Mulcahy has, for the FOURTH time, been passed up for promotion to Captain, making him possibly the most irritable he's been in the entire series. Meanwhile, a Greek soldier and Turkish soldier in the hospital keep getting into fist fights. After they blow Mulcahy off for trying to break up their fight, Mulcahy angrily threatens to beat them up, immediately stopping the fight. I believe his exact words were "I suggest you remember what it says in the good book: love thy neighbor, or I'll punch your lights out!"