Well, no. Women have their own public places and institutions not explicitly because they have their own needs per se, but more so because it is a very bad idea to mix average women with average men in those institutions, for probably obvious reasons, which are chiefly tied to social roles but also biological ones to a lesser extent.
I mean, you could have a public bathroom with all accommodations for both males and females. It still wouldn't be a good idea to mix them generally.
Because, like it or not, gender, being a social identity, heavily shapes the way all of us think, including you and me. It might be sad that it has to be that way, but it is sort of inherent given that humans have had some form of society before we were even truly sapient on the level that we are.