@qork Yeah, I'm in the US too. I can only speak with authority about California where I've been in HR. But here at least, it is universal that women can request to be off work on disability beginning 4 weeks before due date without having any exceptional reason beyond "I'm super pregnant and uncomfortable."
I know that at least 11 other states do this as well where they offer pregnancy disability leave.
Your state may not offer paid leave, but your doctor can still write you out as disabled. You may not get paid for that day, but the point is that your supervisor cannot just say "no". If she does, you can just say "ok, then I'm on FMLA starting now, see ya." You have that right as long as you've worked for your company for at least 12 months.
Your supervisor is probably checking with HR how to handle the pay/policy, but if she's actually considering saying no, she's going to find that's not her decision.