Most Helpful Guy
If he isn't responding then I wouldn't consider it cheating. I think he has the right to read the email before deleting it. Is it the same woman, or multiple women?
I would say that you accidentally saw his email and noticed the emails. Don't tell him you've been spying. Also, stop spying on him. You're violating his privacy. You wouldn't want him reading your emails without your knowledge. Tell him you saw the emails and just ask him about them and tell him it was an accident that you saw them, but what's done is done and they made you uncomfortable. It's what you should have done when you first noticed them. You don't "need to spy on him," you need to talk to him.
If it turns out it's nothing, he's guilty of nothing and your guilty of violating his privacy and his trust.1