In the movie Lady Bird (2017), why did Father Leviatch quit being the director of the plays? Is his sad past being told by one of the girls really true? Why did he and Ladybird's mother meet and talk? Why did he say "Do not tell Ladybird"?
Whether the stories told about his life are true or not (this is never clarified), it's obvious Father Leviatch has deep emotional issues. He sobs suddenly and uncontrolably and apologizez for it in the crying exercise and later goes to see Marion (who, remember, is a psych nurse), where she asks him questions that are typical for those suffering from anxiety or depression ("Do you have a support system? Who do you turn to when you feel this way?") And by the way the scenes are structured, it's indeed implied that this is why he left the play.
Furthermore, when the script introduces his character, it says there's something "depressed about him", and later on in his meeting with Marion:
INT. PSYCH HOSPITAL. CHECK-IN. DAY.
Marion helps a patient fill out a questionnaire. It is Father Leviatch, who looks how he feels - crushed by bottomless despair. This is his struggle.
As for asking Marion not to tell Lady Bird, it's possible he just didn't want to upset her, but I mainly think he just doesn't want everyone to know about his problem. It makes sense: Marion might tell her daughter, she's his student and pretty soon the whole school might find out.