Would it theoretically be possible to engineer a virus that targeted only men or only women?

What about pregnancy?

In principle, yes.

Men have much higher testosterone levels than women and women have much higher estrogen levels than men.
Just make a virus that somehow requires these chemicals to become active. Even though it would technically affect both men and women since both have testosterone and estrogen in their systems, said virus could be designed to have negligible effects below some sort of baseline concentration.

no because gender is a social construct, but you might be able to target a virus for only people with penises

If the goal is to infect a lot of men, target typical male products such as heavy drink's or male hair gel. There's a risk that women might be infected eventually but the majority should be male. This is both cheaper and easier to do.

Yes. Some viruses are gender specific. And IF's its bound to your Y-cromosone, it's for men only

HPV
Done.

why dont we find out?

If you had a virus that does not interrupt testosterone or estrogen, but does interrupt progesterone, you would have a problem.