Ah! An interesting question. Let's assume 'chivalry' is defined as being especially polite to women. It's not, but we'll pretend it does because that's what it means in this era.
So is chivalry sexist? Well, yes. Specifically against males. If you'd do something for a female, that you wouldn't for a male, that's discrimination against males. Sure, some might complain and say it's not, but it really is. Some might say that it doesn't matter, that it's only small things, but it's not.
There are people that are perfectly content with being violent towards men (or just allowing violence to happen to men) but, out of some perverted concept of chivalry, would find the idea of letting that happen to a woman unthinkable. Now imagine you're some guy being beaten up by a 'chivalrous' dude for whatever reason. How cruddy would you feel that your lack of a vagina meant that you could be treated like that in their eyes? It's sick. That's the only word for it, really.
And, for you saying that this all means women have more rights, no it doesn't. It just means that certain men believe they do, and certain women assume most (if not all) men believe they do.
I'll finish by referring to a poll I saw on a forum I saw the other day (it may have been this one, actually) about whether you'd give (or was it risk?) your life to save another. The answers amounted to 'yes', 'no' and 'only if they were female'. Unsurprisingly, a considerable number had picked the last option. I'm just saying, if you were caught in some fiendish deathtrap, wouldn you really want your potential rescuer to base his (or her?) decision on your gender?