What can your employer do if you ask for time off on the day & attend a pre-arranged appointment after he has said you can't go as its too short notice?

If the appointment was pre-arranged, then you should have given your employer notice before the day of the appointment.

Of course your employer cannot make you stay at work.... but they can write you up or terminate you for cause if you fail to adhere to their regulations.

If a employee of mine leave work after being told they can not, they are fired.

I agree, you must follow company policy. What does your employer say, in writing, about proper notice for time off? If I were the employer I would want you to talk to me first about when you could set your appointment.