Apologies, I only read to this post and stopped to reply:
Out of curiosity, would you have a problem with a men's room at a university
Yes, if it were being used for study. Women have historically been excluded from academic contexts, and men are still given more space in classrooms and lectures, so it would be wrong to exacerbate that. The exception, I think, might be on religious grounds or in a context where men are in the minority (eg., nursing). That might work.
or male only sessions at a gym?
Those sound very sensible.
It seems to me that there has been a major historic effort to shut down male spaces, yet we still promote female only space and events.
Please give me examples of 'male spaces' that have been shut down? Spaces where there is a mixed or female equivalent, but no male equivalent? I think perhaps there will be a pattern to what you find when you look.
As this thread demonstrates, female-only events are so far from being promoted, they are in fact under threat. Compare, perhaps, to the Bullingdon club.
I'm curious how this is equality?'
A major tenet of feminism is that women do not yet have equality. I am pleased you are concerned too, albeit late in the day.