Yes, a woman SHOULD act like a lady if she wants to be treated like one. Though society doesn't really hold women responsible for their actions, so she'll probably be treated like a lady anyway.
Part of being a gentleman is treating women like ladies regardless of whether they deserve it or not. This is not true in reverse. Women have no social obligation to treat men as gentlemen if they aren't.
Since it's considered ungentlemanly bordering on misogynistic for a man to criticize a woman's behavior, women are only really policed by other women.
This post was edited by Zeitgeist at July 18, 2017 6:59 AM MDT