I don't see what the problem is. If an employer is expected to pay us for time off, then they are well within their legal rights to ask for formal proof.
I've given a copy of every hospital letter in. I didn't with my 1st pregnancy, as that boss always took my word for it. This boss wants proof and formal records kept. Fine. No skin off my nose. I give photocopies of mw appt that are in the notes. No idea why people think this is an unreasonable thing to be asked. I agree it's not nice to feel as though you aren't being believed - I have thought "when have I been a shirker or lied in work? What cause does she have *not* to take my word for it?". But it is their right, so I just give the paperwork.
Edited to add:
Meant to say...it is absolutely NOT your boss' business what the appts are for. Your medical info is private and confidential. All she has the right to ask for is proof of an appt. When she digs further, just say "I'd rather not go into that, if you don't mind".