Even though all the Armed Forces are subject to the UCMJ, each service has a fair degree of levity in how it enforces it in order to allow commanders to establish a command climate that suits their needs and leadership styles.
I can only speak from my experience in the Army. Service members are servicemembers 24/7 and are subject to their respective service regulations, command directives and the UCMJ even when on leave or pass. We expect our servicemembers to be active and stay in shape, but where do you draw the line with regards to risk? Should an SF trooper or member of the Golden Knights be charged if they get hurt while participating in a civilian free fall club? It would also depend on whether or not the USAF Captain's unit (if he's in Europe) had given a safety brief or command policy forbidding participation in the running of the bulls.
There are a lot of variables. In the Army of my day, I would suspect that, had he not been violating any direct orders or local regulations by participating, a line-of-duty investigation would have been initiated and he may have ended up paying for some or all of his medical treatment if it was believed it was the result of his own negligence.
Thanks, you would think that they will now advise against this then, so he might be the last one to run without facing an article 15 if he gets hurt.