In the US, "on leave" usually refers to military leave, when soldiers are permitted to be away from their units on their own.
The kind of leave from work you're talking about would be "a leave of absence" (paid or unpaid), usually for an extended period of time. "maternity leave" "sick leave" from work are also well-known terms.
American, while it's true we wouldn't use "on holiday", and while "on leave" might be a technically correct translation, most Americans would think it was a bit weird to hear. Normally we'd just say "on Vacation". "On leave" is typically only used for military personal when they are allowed to leave base.
@Sam278989 - We use multiple prepositions in English, too - "He will be on vacation in September". In Russian, when you use a preposition, you have to use the appropriate case for the following adjectives + nouns. Отпуск and months use the preposition B + prepositional case.