I think personally that this guy does not obviously value your friendship as much as you do as he is so eager to 'step-over' or compete with you for female attention. Is this a friendly rivalry, such as "Bet you I can get with her first." but there is no ill feeling? Or is this competition something that has developed on its own accord?
I'm just curious, but you said that your friendship developed out of your mutal love for climbing, so I'm assuming that perhaps the friendship itself revolves around competing against one another, or would you disagree?
Personally, with my friends, especially the ones I consider close friends or there abouts, we would never step over each other for the attention of a girl. I have however met people who can be very condesending towards me, which is something I find intolerable and I get very confrontational about it. Recently a friend of mine has begun to explore some interest in a girl we both met, which originally he thought I may be interesting in, but as I decided she was annoying he concluded that she was fair game. I've only known the bloke about a week so far, but we've already got mutual respect (or at least I hope so). Your situation probably just shows this man's lack of character.
If I was in your exact situation Blueski, what I would do would be to not change my attitude at all. Distancing yourself may only mean that he'll do it again in the future. Your friendship is still what I'd consider new, so I think that you're still testing the water in most cases. The next time your friend becomes snide, or arrogant towards you; put him in his place.
I've found that saying "Who the **** do you think you're talking to?" often works wonders.