I am sorry to hear that your friend fell and hurt himself. There are two parts to your dilemma: First, who should pay the medical bills? Second, what can you do about the strain in your relationship with your Fallen Friend (FF)?
1. You indicate that FF has medical insurance so it is clear that bills will be paid by that authority, the amount conditional on his type of coverage. However, you say that FF has cast blame for his injury on a third party and seeks reimbursement from that party. This action brings forward an ethical issue because you think that the third party is not liable. If your analysis is correct, then it seems to me that FF is playing on both sides of the street. This is hard to do, unless he can avoid traffic when running from one side of the street to the other. Usually, the person gets mowed down by cars.
2. Now to the second part of your dilemma which is how to alleviate the strain in your relationship with FF because you disagree with his ethical choice. You must be straightforward with him. If it is difficult to tell him face-to-face that his decision to seek reimbursement from the third party is immoral, then put your analysis in a gently written personal note, not an email or a phone call. Write it once, leave it, then read it and revise it. I can tell that your goal is to preserve your friendship. So after you express your opinion, either in person or in a hand-written note, call and invite him for a quiet evening in your home, where you can both listen and talk to each other with no external stressors. If he does not live near you, then perhaps you can arrange a time for a long talk via phone.
Good luck, and please let me know how you and FF are doing.
Dr. Joan’s Harmony Key: He who plays fair shines light into dark corners.