Is spying on him ending my relationship?

Ending the spying won't make the problem go away. The spying isn't the problem. The problem has to do you not trusting him (if you did, you wouldn't have started spying on him in the first place). If you stop, the unsatisfied curiosity will just fester, making you harder to deal with. Whether your distrust is justified or not, you'd do well to just end things him him ASAP, since the whole relationship is doomed to failure (a relationship can't survive without trust).