Hmm, no we don't say "wet on himself" in the US either. Actually I had thought it might be a UK thing, but I guess not.
Yes, "piss" is vulgar, and wouldn't usually be said on network TV, but I wouldn't say it's as strong as "shit". For example, I can't imagine it being censored or bleeped out, while "shit" certainly would. That's just in my experience, though.