"Welcome Home, Part 1" is full of moments like this, as Richie and Ralph come home from their stint in the army. One notable moment is in which Richie, Fonzie, Potsie and Ralph sit at their usual booth at Arnold's. At first there's an uncomfortable where they have nothing to say to each other. Ralph goes over to the piano and plays the opening notes of "Blueberry Hill" Richie's signature song. Richie starts singing, then Potsie and Fonzie join in and they all gather around the piano, smiling and singing the rest of the song. (This would also be the last scene in which all four characters appear together.)
Richie jumps into frame and immediately the audience begins screaming and cheering - this goes on for minutes on end as he hugs every member of his family.
The episode "Richie Almost Dies" is this, especially the scene with Fonzie making a deal with God ("You do this for me... And I'll owe you one.") at Richie's bedside, before completely breaking down in tears. There's a meta element to it, as well: The entire reason the episode was made was that the producer had received a letter from counselors who were having trouble with abused children. The kids wouldn't express their emotions - they didn't want to cry. They all loved Fonzie for being so tough, and the counselors thought that seeing him cry might help them to realize that it was an okay thing to do. The fact that a popular tv show cared enough to respond to this sort of request in such a big way is quite touching, no matter what you might think of the episode itself.
In "They Shoot Fonzies, Don't They?", Fonzie gets annoyed on Joanie's behalf when the head of the cheerleading squad, Jill, purposely keeps her out. Jill makes a bet with Joanie - if she and her partner can outlast her and her brother in an upcoming dance marathon, she can join. Fonzie agrees to the job. Then, the night of the dance, his motorcycle gives out and he has to push it several miles simply to get to the Cunninghams' house. In spite of his ordeal, Fonzie can't say no to Joanie; as he later explains, "This dance means so much to her." All together now...awww.
Fonzie's relationship with the Cunninghams in general is this. Whether it's expressed as dramatically as in "Richie Almost Dies" or more quietly, such as with his determination to keep on his feet during a dance contest for Joanie, it really reminds us that the town's favorite cool guy has a heart under all that leather and tough talk.