I am a bit confused about that too. However I don't think the hadith says to apologize. What I heard was that we should not be on such non-speaking terms with another muslim, so as not to answer their Salam for more than 3 days.
Anybody have the text of the hadith?
As I recall, this applies to general situations when either of the parties should try to resolve the dispute; in special circumstances - which can be many - of course common sense exceptions exist.
I really cannot remember anything more on the nature of the hadith.