good day

Is 'good day' a correct formal greeting like 'good morning' and 'good evening'?

yes -- perhaps more common in some places than others.

I always equate "good morning" and "good evening" with opening greetings but "good afternoon," "good night," and "good day" with farewells. Good morning can be informal, but the rest are formal where I live.

I've thought saying "Good day" is a rather rude way to stop an unwanted conversation implying "Don't bother me any more".