Yeah, sometimes you just know when someone is bullshitting you though, and if they won't admit it you have little recourse but to satisfy yourself. You should have already made the decision to end it regardless of what you find by that point though.
Well, yeah. What's the point of confirming your suspicions when doing so violates the relationship in and of itself? What would have happened if he had looked at her email and found nothing? Then he would have been guilty of a complete asshole move and by doing it would have been faced with the option of either lying to her by omission (same thing she's guilty of-violating trust and not admitting it) or telling her, which would have ended the relationship anyway (assuming this girl has any self-respect at all).
He could have just taken the high road, told her that he knew that she wasn't disclosing everything and ended it based on the fact that the relationship clearly sucked anyway.
Well yeah, but if it's over anyway you might as well find out for sure. If you are convinced enough to do that, assuming it's not the sort of thing that you do at the drop of a hat, then presumably you are past the point of caring about the privacy of someone you think is screwing you over.
The fact that he reached this point now doesn't necessarily mean he did anything to push the relationship to that point. Or maybe he did, with only one side of the story available we'll never know.