Here's a fun scavenger hunt for you: see where and how often the word "woman" shows up in The Great Gatsby. (Helpful hint: this online text is searchable.) We'll give you a hint: it's mostly in reference to lower class women, like Myrtle or some of the servants. Upper class women are "girls," like the "men and girls" who wander around Gatsby's garden (3.1). That doesn't quite tell you all you need to know about gender in The Great Gatsby, but it tells you a lot: Fitzgerald is no feminist, and neither, apparently, is Nick.
In The Great Gatsby, men and women don't make each other better; they just make each other worse. So much for chivalry.
Women in The Great Gatsby are mostly there to entice and subvert men. Without women messing things up, life would be a lot better.