Is it correct to include a name right after a he's or she's?
It would define fully written in: ''He is John'' or ''She is Maria'' but it somehow just sounds wrong or impolite to me.
It's perfectly correct in informal use or when speaking. When writing, especially formally, it is better to expand it.
In informal speech this is correct, however it might be more polite to say "this is John" instead of "he is", or "this is my friend John" if you are first introducing someone.
To me, by saying "he is" it almost makes it seem like you are not good friends / you do not have a good relationship with each other.