Your question is a big, marvelous one, Mia, and I've been thinking about it.
But all my thinking so far has come to naught.
I've found no way to separate gender from need and procreation and sex.
I think that because we are gendered we do need each other--there's no way out of it until science figures out a way that men can do it (procreate,have sex,perpetrate rape, use as scapegoats) all by themselves, without women involved.  What happens when men don't need women anymore?

This isn't any kind of answer at all and certainly not the kind you're looking for
and I don't mean to irritate the very intelligent gentlemen who populate this site.
'Is gender a way of reminding us that in a deep sense we need each other, or a false set of roles leading to imagined incompleteness?'
Gender doesn't remind me of anything.  It exists, it's a fact, it's not something I decide to leave at home when I go out.

Am I not understanding your question?  It's like I'm being asked how I'd feel if I lived without air.

As Linda says this is a very broad and complex question. For starters we could define our terms: what exactly is “gender”? I’m not trying to be disputatious but this is actually a strongly contested term in some circles. Gender can be differentiated from sex, in that sex is determined by biology while gender is arguably a constructed identity. Which leads ineluctably to the next question: if gender is at least partly a question of identity, is it entirely socially determined or it partly based on innate characteristics  (the classic “nature vs. nurture” debate)? To put things very crudely: if we gave boys dolls to play with and girls toy trucks would the boys grow up to exhibit more “feminine” characteristics and the girls more “masculine” ones? If there isn’t even a consensus on what gender is how can we possibly answer a question like “In what sense do you think we need people of the other gender to be complete?”

Although I find these kind of discussions interesting I personally think within the context of this question they are somewhat gratuitous, because I reject the question. That is, I don’t think it’s healthy to feel that you are incomplete without someone else (regardless of their gender!) If you aren’t happy as an individual how can you be happy as part of a pair? Granted, we are constantly bombarded with social messages that tell us we can only really be happy in a relationship, but I think it’s important to recognize that messaging for what it is and to question it.