In the abstract, any one or more of thousands of things might happen.
Confessions don't get prosecuted. People get prosecuted. Whether the police will investigate or the prosecutor might prosecute would depend on the relevant facts and circumstances, none of which appear in your post.
The confession is evidence, and the victim is a witness.
It is hearsay. Cal. Evid. Code section 1200. However, it is not excluded from evidence because of the party admission exception. Evid. Code section 1220, and the unavailability of the declarant is not an issue. Of course, it is correct that the issue of hearsay won't come up until and unless charges actually get filed.

