Most of the time, you'd just make eye contact if that. granted, i'm talking about small bumps and things that Americans apologize for daily, but are a normal part of life. Things change a little if you actually get someone to lose balance or fall down. "unnskyld meg" - a form of "excuse me" - same thing someone says if they want me to move. More importantly, make sure the other person is OK. Help them pick up stuff if dropped. Mostly, it just more seems like that sort of thing isn't as big of a deal here because it is part of life, especially on crowded buses.
Don't get me wrong, I find society really nice on the whole. If I fall down on the ice, folks make sure to ask if I'm OK. People help children (and children stop random folks to talk... they are on public buses at school age by themselves). Most people have been really patient with the language barrier - more so than I saw in the US - and people are happy I try instead of being upset that I have an accent. And granted, I was taught some of this from my Norwegian spouse and through language/civics classes required for my immigration.
I honestly hear apologies and the like more from other immigrants, and Americans seem overly polite and fake happy/apologetic in comparison.