The OP has not asked for suggestions on how to deal with "dad fallout" or stop dad from talking to her boss, but how exactly she should talk to her boss/manager herself....
I think there have been some response - but the only real answer is that she states her plans and doesn't back down.
"I have an doctor appointment at 3 today" "I'm sorry, I need you to work at 3 today" "I'm afraid that won't be possible"
"I have a headache and I need to leave" "I can't let you leave, take an aspirin" "I'm afraid that won't be possible"
The boss and crony may never believe the seriousness of the OPs condition. They may always believe she is exaggerating or faking. There is only so much the OP can do about that - her first responsibility is to herself and her own health.
I think this phrasing might get someone fired and not just OP.
"I've leaving to go the hospital/urgent care/to seek immediate medical assistance/advise" not "I have a headache and am going home." OP I am making an assumption that you will seek medical attention if you have to leave during a shift even if that means calling the doctor's office and having the nurse say "yes you must do x y z ( which includes leaving work). IF you are not going to be seeking medical attention in any form " I am very ill and must go home now" , "you can't take a pill. " "I'm sorry but I am too ill to continue working I must go home now"
As soon as you know the appointment time " I have a cardiologist appointment at x time day. I will need to be off y time (include travel and wait time at the office" , "you have to work that day" , " This was the only appointment available within 30,60,90 days. I must see my cardiologist, I can not reschedule"
I don't think "that wont be possible " is an appropriate response to being told to do something within a persons normal job duties. Unless its to a question including "is this possible ?". "That wont be possible", is a way to say no without explaining or debating further.
. OP is not open to debate, but when asking for an exception to your normal job some explanation is in order such as "No appointments available on my day off.". In this case in particular this manager already seems hostile , "that wont be possible" I think would be seen as confrontational.
Try to work with a different manager whenever possible , keep them as informed as reasonably possible. ie " I have been trying to see my cardiologist but there is a 180 day wait for an appointment. I will be given the first chance if the office has a cancellation before then, so I may have an appointment with short notice in the near future"