Wolf, it isn’t rude not to greet others THAT U HAVE NO CLUE WHO THEY ARE AND U JUST PASS THEM BY ON THE STREET.
You don’t have to shout. And yes, I believe it is rude. Would you walk by someone of the same gender on Shabbos and not say “Good Shabbos?”
Does your Rav greet your wife on the street? Does his Rebbetzin greet you? It isn’t a mitzvah in theleast bit.
In fact, my rav has made the point publicly that it is common courtesy and there is nothing wrong with saying “Good morning” or “Good Shabbos” to someone of the opposite gender.
The Wolf