If we define being gentlemanly as chivalry, Websters dictionary tells us: the qualities of the ideal knight : chivalrous conduct. Knowing that to be the origins, a more recent meaning of the definition would be : gallant or distinguished gentlemen. Chivalry is more than treating women differently; it's a moral code, a guideline by which one should live his life, treating everyone you meet correctly. Being chivalrous to women specifically, however, clearly falls under the definition of sexism: a prejudice or discrimination based on sex; especially : discrimination against women.
As far as I can tell, being a gentleman is not to be guilty of sexism; however, if chivalry is only practiced when women are around, I would say it's definitely sexist.
If that's a bad thing is a different topic entirely. While I can see why some people would see this benevolent sexism as a bad thing, for me, it's mostly how I've been raised: to be polite to women (and men), and for me personally, not a single sexist thought goes through my head when I offer a lady my jacket, or hold the door. One could argue that even without intent this reinforces the stereotype of the woman needing manly assistance, but take my word for it, no harm is intended. (In my case, anyways)
As far as I can tell, being a gentleman is not to be guilty of sexism; however, if chivalry is only practiced when women are around, I would say it's definitely sexist.
If that's a bad thing is a different topic entirely. While I can see why some people would see this benevolent sexism as a bad thing, for me, it's mostly how I've been raised: to be polite to women (and men), and for me personally, not a single sexist thought goes through my head when I offer a lady my jacket, or hold the door. One could argue that even without intent this reinforces the stereotype of the woman needing manly assistance, but take my word for it, no harm is intended. (In my case, anyways)