False, sometimes no prosecution is lack of evidence, unwilling to testify by witness, maybe never reported to actual police, and finally a settlement can happen.
No prosecution = Roethlisberger righly accused. As I recall, the DA said something very close to, "My not indicting indicates I do not think I have a good chance of proving guilt. It does not mean I believe the alleged perp is innocent".
If he had been charged, convicted and Iowa had allowed him to remain and play, to a lesser degree the University would have been thrown in with PSU for putting football above the safety of students. At least that is how I would see it.