Are you in the U.S.? I thought FMLA is for extended period of absence, like a long-term illness or maternity leave. It’s to protect workers from losing their jobs. I think it’s worth googling to see what the law actually states.
Specifically does he mean for you to take an unpaid day off? Do you have sick days or vacation days you can use up first? I’ve had employers who want you to “make up the time” missed for doctor’s visits, etc., but it’s nicer when they can just let you take the hour or two, especially if you’re salaried and you’ve already proven yourself to be a good worker.
As for the ADA, I think it’s there to protect people with disabilities from discrimination, not to help employers figure out who on their payroll has disorders. Even if someone has a disability, though, that doesn’t mean they’re exempt from their employer’s policies. Good luck!