This is one of the enduring mysteries of the universe to me: why do we have separate male and female toilets? From an efficiency perspective it is clearly massively suboptimal. There is regularly an empty stall (or two) in the men's toilets while there is a queue for the ladies. Or both genders end up having to walk unnecessary distances to use the facilities (as actually happens in our maths building). One reason might be that women don't want to be in the same room as urinals, but that's easy to solve: just have a separate room for urinals.
I think this is a massive example of the grandfather fallacy. If we lived in a world in which there had never existed the sort of prudishness which makes separate male and female toilets a necessity (I mean, you get a lock on the stall door!), I don't think there would be any proponents whatever for a move to introduce them. As it is, there are a limited number of people who manage to come up with plausible-ish reasons to uphold the status quo whenever I bring the subject up, but I simply don't believe that any of those people would be campaigning for toilet segregation if we lived in a world in which integrated toilets were the norm.