In Avatar (and in most of the films that is said to have inspired Avatar) the protagonist that crosses over from the western civilisation into another culture is a man. One of the reasons (or perhaps the main reason) to this crossing over is his love to a native woman. The other way around, that a western woman falls in love with a native man, and crosses over into his culture, is much more unusual in films. But let us just play with the thought in the case of Avatar. Just imagine if the Jake character had been a woman and the Neytiri character had been a man? Would people have viewed Avatar in the same way, and would it have become as popular?
By the way, if you want to see a film about a Euro American woman, falling into love with a native (Native American) man and crossing over into his culture there is
Stolen Women, Captured Hearts from 1997. It can be seen in its entirety on Youtube.
Stolen Women: Captured Hearts - Wikipedia, the free encyclopedia