I don't know about in the states, but where I live if I walked through a door without allowing a woman to go through first (especially if she is going in the opposite direction and the door is only wide enough for one person at a time), I would get a very cold look indeed. I agree with Seyorni in that chivalry does originate from the "myth" of female helplessness, but I doubt any sane women see it that way now. It's just a harmless tradition and most of the women I know would be appalled if a man didn't offer her his seat on the bus or something.
Interestingly, I saw a very good TV program this week about old inventions that were patented but never released. One of the patents was from the early 1900s and was for a clockwork, automatic "hat tipper", which would raise a mans hat automatically when his hands were too full to do it himself (as is the tradition in England). A scientist decided to take the blueprints and make the contraption as an experiment. He went to Harrogate, Yorkshire, and tried it out on the local women. He asked them "would this be a suitable replacement for a proper hat tipping if a man's hands were full?", and the reply 90% of the time was "no, he should put his bags down and do it properly, it's just good manners".
Try telling the women of Harrogate that chivalry is bad! =P