You have dozens of answers already, but here's a new one, just to add my opinion to the list
In two sections:
1, sexism. Yeah, treating a girl differently than a boy of the same status is technically sexist. The act of helping her out was charitable. She doesn't like being seen as a charity case. You were trying to be nice, but clearly, there is a misperception of women as requiring, or worse, DESERVING your charity solely for being women. Though this can be seen as either discriminating against ("he doesn't think I can do it myself, what an asshole")or in favour of ("he thinks women deserve better treatment than men, that makes me feel superior.") women. Though I've personally gotten in trouble socially for both holding the door for a girl AND not doing so. In my experience, it depends on how much of a jerk the girl your dealing with is.
2, douchebag. This is the part where she was totally out to lunch XD! Doing nice things for people without any expectation of a return favour is honorable, regardless of your other reasons. Your not a douchebag buddy, just a little misinformed, and a little old fashioned. The only thing you should fix is that you should be helping out everybody, because men are people too.
Keep being a nice guy, just, be a nice guy to everybody, and don't worry about the bitches, because they'll fault you for their own perceived shortcomings no matter WHAT you do XD. Good luck my man