Is the bold part correct?

It is acceptable (although the common version of this locution is "met with an unfortunate accident"). However, the rest of the sentence is rather inept. The dependent clause ". . . where a blood vessel in his nose ruptured and had to be hospitalized." reads as though either his nose or the blood vessel had to be hospitalized. The use of "where" is awkward also. The dependent clause would be better written as: " . . . when a blood vessel in his nose ruptured and he he had to be hospitalized."

Thanks, Setanta, for the corrections.

I agree with what you said.

To correct my response, the second "he" in the final sentence is superfluous.

Thanks, Setanta.
I'm aware that the second "he" is not needed.

unfortunate circumstance ...

When I read this sentence in my head, it sounds like something a mafia enforcer would say at a police interview or to a judge while he's on the witness stand.

It was in a dark alley where he met with an unfortunate incident. So very unfortunate of him.