I posted this on another skye/ryan thread but it's exactly what I want to say here:
I think Ryan took Skye at face value at the beginning of this game (bimbo, dumb, shallow), and particularly after joining the 'older', 'more intelligent' group, feels he can't now redeem his initial impressions - which is why I was so impressed with Dave telling Skye he never gave her credit for how mature she was.
Ryan sees himself as superior to Skye (in intelligence, capability, "worthiness"), and feels frustrated that she is so popular with the public/other housemates and is a real threat to him. Now he's just throwing in the towel because he knows he has a real shot of going next week, and "isn't taking Skye's crap anymore", aka he is knowingly making himself more unpopular and more likely to go home, but would rather do this - he would rather take his frustration of Skye 'beating him' out on her than man up and admit he had her wrong.
He's trying to make her seem immature and dumb in the argument, when really, I think she's onto his game and she's so over it that she was willing to say "fuck off". That just gave him more fuel to get mad at her for but everything he said in that argument was an eye roll and everything she said indicated she was aware of his position. Go Skye