“rationalfrummie” – actually separate entrances or clincs for men and women would be quite legal in the United States – unlike some countries, virtually all hospital rooms are single sex — bathrooms and changing rooms in public facilities are always single sex – the standard of religious accomodation is that an employer or service provider can not be ordered to accomodate religious practices if it creates a “de minimis” cost, but that such accomodation is allowed and is generally considered praiseworthy — when a university agreed to accomodate Muslims with separate swimming, the objection was based rather clearly on hatred of Islam, not on any gender based theories. Remember Israel is a country reknown for all sorts of things of the type we can’t discuss here (such as “recruiting” young women from other countries to do such things for pay that we can’t talk about in proper company – and advertsing the fact to attract tourists). It’s not about status of women – it’s about hatred of Bnei Torah.
it should be understand that the rules against gender separation reflect not a desire for sexual equality but are based solely on hatred and bigotry against frum Jews – based on what we’ve seen elsewhere we can expect that this bigotry will probably evolve into much stronger measures to reduce the number of hareidim (perhaps following the proverbial czarist formula – assimilate a third, kill a third, and convince a third to leave the country).