Let me be clear: You
can of course refuse to serve someone for any reason that isn't illegal. If somebody is causing trouble and is drunk, of course you can throw them out. What I'm saying is that it's not a "right to refuse service", per se. It's just what is and isn't illegal.
The reason I specify that distinction is because so many people think there
is some sacred store owner's right to kick somebody out, especially people in the United States who see "we reserve the right to refuse service to anyone" signs and assume that if someone claims to be "reserving" such a right, it must surely exist. But it doesn't, not in that general form anyway. The government can and does prevent you from refusing service in many cases.
Of course, there's a big difference between what should be true and what is true. You can, of course, advocate that there
should be such a right guaranteed to business owners, or maybe that there's a "moral right", even if it's not recognized by the government. I'm just being such a hardass about it in hopes of dispelling the very pervasive myth about the actual rights that business owners have. I'm sorry if I'm being
too overbearing -- I tend to come on very strong when I'm trying to stress a point of fact, rather than state an opinion.
I guess gyms are either private clubs or public establishments depending on the particular argument going on at the time.
Go to Google and type in male-only gym. Notice anything? There don't seem to be many, if any at all. Google will helpfully pipe up that you may have misspelled female.
Yeah, I actually thought about mentioning gyms, but... I wasn't sure what to say constructively. I honestly don't know how they're generally treated legally. I'd assume a "women's health club" would be fine, but is a "men's health club"? You might be able to justify it (after all, men and women are different in some matters of health), but it would surely be controversial.