This issue has come up in the past and as I recall one Austrian airline had segregated bathrooms for a short while. The pros and the cons:
Pro: Men leave bathrooms messy, toilet seats up and sinks full of soapy water ( thank heavens for the evidence they wash hands). We will not even talk about the floor situation.
Con: Women spend more time with make up and hair in the airplane bathroom. Women also use Purell more often than men.

On many non US planes, cabin crew actually cleans the toilet areas so men vs women is not an issue But in the US we believe, when a bathroom is left a mess, we should photo it and request it be cleaned, restocked or closed as “”Out of Order Due to a Messy Passenger on this Flight. ” And Purell should be available in a dispenser either in the bathroom or outside its door.

What do you think?