I've had to cancel three nonrefundable trips in the last 6 months due to two separate medical issues, so I have pretty recent experience.

UA will only truly refund (to your original form of payment) nonrefundable tickets when your doctor's note indicates that you will be unable to travel for the forseeable future (basically, the validity period of the ticket for which you are seeking a refund). OP, your situation probably qualifies as this. You're still out the $50 processing fee that they charge.

If you were unable to travel because of medical issues that have since resolved or at least changed such that you are able to travel again, you can get a refund of any change fees (minus a $50 processing charge) associated with changing to a ticket you are actually able to use, but they usually aren't willing to refund the ticket itself. UA refunded all of my change fees, including a $500 change fee on an international business class ticket (minus the $50 processing fee).

Finally, you should check with your credit card as a situation like this may qualify for a trip cancellation insurance claim if your credit card offers that protection.

For what it's worth, I had to cancel two trips on AA for the same reason, and they weren't even willing to waive the change fees.