There have been experiments with that, with parents who have done everything they can to isolate out cultural factors, influences, you name it, and what they seem to indicate is that certain things are not learned behavior. The whole nature versus nurture thing has been argued out ad hominen, and those arguing the purist point have been blown out of the water time and again. Among other things, I recall reading something several years ago where they did cross cultural studies of societies, including isolated tribes who had little to no contact with the modern world or the rest of the world, and they found there were certain things that came up time and again across cultures. For example, the kind of play little boys and girls tended to be attracted to. The studies authors were saying that when you see the same thing time and again,in cultures that are isolated or part of the modern world, that it is a strong indication it is inate.
There are obviously things that are cultural, the concept that girls make themselves pretty to compete for the attention of partners, for example, is cultural (in some cultures, men do that). Human beings are not clean slates, though, we have instincts, driven by millions of years of evolution, and how for example hormones affect things are very, very complicated (estrogen and testosterone are both powerful things, they are a lot more than why a man has hair on his body and a woman develops breasts), and that plays into it. If the theory that gender is wired into the brain via the hormone washes that a fetus undergoes (and as I wrote in another post, there is evidence to support this from experiments), then the answer would be no. How we express that gender identity would obviously be very different if we could isolate out the cultural stuff, the learned behaviors, but I donāt think it would affect the core identity. Put it this way, our core gender identity is so intrinsic to who we are we really donāt notice it, If I tomorrow I got rid of the label of male and female, wiped it out of peopleās memories, and simply called everyone āhey youā, inside I think people would still feel and act differently.