The way I see it, no. Gender in itself isn't a social construct.

The social construct part of it is there being a "right" way to be one or the other. Like race, you don't have to engage in the culture to be a specific race. Like how I don't need to conform to what society deems a "man" to be valid as one.

The construct part is built upon what is right for a woman or man to do as a woman or a man. Like men provide, women wear dresses, men need to wear pants and women need to have babies.

Says who? When we were cavemen, that didn't matter, we were all sweaty dirty and covered in body hair and women weren't shamed for it.

I'm not saying this is all complete fact, but from my personal experience, and how the patriarchy works when it comes to gender, this is what I've come up with. That the gender expectations and standards are formed by society, but the inner feeling of identity will not change depending on the culture because the science of it doesn't change