Techman,
Not sure I agree with your comment that a "medical cannot be on a day off". If the company roster the medical then you have a valid point. Usually you are simply required to obtain a medical at your convenience. In this case you are not being rostered to obtain same, but have to show evidence by a certain date. In this case I dont believe that, although it is a company requirement, it counts as "duty". Renewing a license or obtaining a passport would fall into the same category.
My point being that unless the company required you to report somewhere on a specific date it could not be classed as duty.