No, you couldn't use 'case' here, but you could use 'occasion'. 'Case' refers to a more complex situation. It is similar to the words 'circumstances', 'conditions' or 'state of affairs'.

"I'll never forget this incident" is better. "Occasion" has multiple meanings, one of which sort of works here. You could say, I suppose, "I'll never forget the occasion where I stumbled and fell over on the first day of school, provoking laughter from all who saw the incident." "I'll never forget the time where I stumbled and fell over on the first day of school, provoking laughter from all who saw the incident." (using "time" is simpler and uses a 50 cent word instead of the $2 one). I really have trouble using "case" in this sentence. Consider this... "I have a tendency to do stupid things when I'm nervous. A case in point was the time that I met a pretty girl on the first day of school and got so flustered that I stumbled and fell over, provoking laughter from everyone including the girl in question." "Case" is another word that has many meanings. I could explain some of them but that would get away from the original question so I'll stop here.