The west is just mid level in terms of its segregation.
Some cultures there's not much segregation and people even bathe mixed or share toilet facilities. Some like in the Middle East have it cranked to 11. I think we're in the center, still wanting male or female spaces, but not wanting a truly segregated culture where men and women can't freely mix. We have unisex toilets as well, though, maybe in future there will be more of that. Nobody tends to have segregation at home - just around strangers in public which makes me think it's largely because females have feared for safety or decency while using public bogs historically, and still do tend to prefer female-only settings for peeing and changing etc.
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Blame the British.
Pretty sure you'll find gender segregation or roles to some degree in every culture, including ones the British had nothing to do with founding. Although as a Brit I'm flattered we still consistently spring to mind as the best moustache-twirling villains ever.
Apparently Romans and ancient Greeks had separate bathing facilities, and toilets sometimes, as did Muslims, so it was going on long before us.