Ugh... So, yeah, it's never been done - if it had, this would be open and shut.
Human women often give birth prone such that the force of gravity is perpendicular to the direction they're pushing the baby. Therefore, there's no reason to assume that [most] women aren't strong enough to give birth in zero-g.
Women from a variety of species (including some humans) give birth under water where the forces of buoyancy and gravity are equal and opposite, so there's still no reason to suspect that gravity is a requirement for birth.
One of the long term effects of weightlessness is that our bones become weaker. It's possible (though, highly unlikely because the baby doesn't go between hips - it goes through cartilage and other tissues) that this would mean that giving birth necessitated a risk of breaking one's hips. It's also possible that the bones could become more pliable, either because of the sometime inverse relationship between strength and plasticity, or because of a drug/supplement administered during gestation.
If birth without undue risk of catastrophic injury was not possible, there's always Cesarean section, see bulldogs.
In the weeks to hours before birth, the baby resides lower in the abdomen - a transition that physicians call "lightening". A hypothesis has been advanced that a baby won't drop without gravity. On the surface, it seems to have merit, but babies drop while expecting mothers are in a variety of positions. In addition, dolphins appear to have a similar phase, so at the very least, mammal babies can drop without gravity. Earth women have exercises and movements which hypothetically move the baby into position, but while that research appears to be in its infancy, there's no reason to avoid talking about the zero-g versions of the same exercises.
Humans carry babies in their wombs surrounded by hydro-static pressure. Since Earth women do yoga, swim and sleep, it's pretty clear that their orientation does not effect their ability to bring a baby to term. For women to need gravity to bring babies to term would require proof.
Again, nobody wants to admit to having had sex in space, but humans are endlessly creative when it comes to this. I will not be adding a link.
Nature will find a way. There are few things that cause rapid evolution in "natural" species, but illnesses and breeding problems are at the top of that list. Nature is neither fair, nor kind about this, so there would be some cruel generations where natural selection takes care of this problem. That could have it's own consequences because our powerful brains are related to the long gestation times. The other option would be to guide evolution with Genetic Engineering, but that tech is probably a long way away.
If you want it to be an issue, you can, but I wouldn't bat an eye if a fiction writer told me that zero-g birth was commonplace and about as catastrophic as Earth birth.