The only reason combat rules have such primacy is because of the genre conventions on which most roleplaying games are based.
If roleplaying games had been based on Jane Austen novels instead of sword & sorcery short stories mingled with a handful of high fantasy novels, they may not have had combat rules at all.
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I think you do the question a slight disservice by binning things into either combat or social interaction. There's plenty of other types of situations that can come up, too. Social interaction is tricky, though... in a roleplaying environment, you can, well, just roleplay a large chunk of the social interaction without needing any mechanics to do so, and in fact I'd wager most players see that as desireable, at least to some degree.