I am talking about situations when you want to greet someone, without stopping to talk to them. In Russian, people usually say "добрый день" which means "hello." This is a greeting, not a valediction.
In English, it seems to be opposite. Can I say "hi," or should I say something like "have a nice day" or "bye" when I am passing them?
In English, it seems to be opposite.
I am not sure where you heard that, but it is not true. In English, we only greet people with, well, greetings. That's why they're called greetings.