Thanks for your question. My name is XXXXX XXXXX I have over 10 years of experience helping people with relationship problems.
I certainly respect your willingness to own up to your mistakes. After seeing those texts I can see how it would be hard to avoid continuing to snoop despite it being a breach of trust. Most people would consider his behavior to be inexcusable, especially in a committed relationship, and regardless of how you found out about it. If he is flirting with other people and making arrangements to see them, then it doesn't matter if 'nothing came of it,' because he knows that he is hiding it from you and that you would be upset by his behavior if he was honest with you about it.
You absolutely have a right to be upset, and if you decide to break up with him I would assume that it is because he has violated your trust. He could make the argument that you have done the same thing, and he may be correct, although looking at someone's phone would usually not be considered as significant as flirting and trying to meet up with other girls while in a committed relationship. Relationships have to be based on trust in order to stay healthy, and if this incident has caused you to be unable to trust him or vice versa, then you may have to consider ending it.
Overall it also sounds like you had some legitimate reasons to be suspicious and snoop in the first place. That may not justify it, but since you have actually found evidence of what you were afraid of in the first place, he needs to hold himself accountable for his actions too. I agree with your statement about his behavior undermining your relationship, and if he continues to hold you responsible for his behavior that would also seem like a fair reason to end the relationship unless he is also willing to take some responsibility for his part. I definitely wish you the best with all of this, and if there's anything else I can do to help just let me know.