Did they ever show Fonzie punch someone on the show? A: Yes but it was only once.
Only because it was out of necessity so it was needed as self defense for Chachi. As mentioned before, they implied and off camera had Fonz punching and beating guys up but never showed it because this is a family show. BUT in.....
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Season 11 Episode 235 "Glove Story" Tired of Fonzie treating him as a kid, Chachi enters a boxing tournament to try to earn some respect. But when Chachi breaks his hand practicing, he tries to keep it a secret. He is trapepd in the ring being beaten up unmerciful, and even after the towel is thrown in the guy continues to pound him. Fonz has to step in and this is the ONLY time you will see Fonzie ON SCREEN hitting someone.