I agree, this sounds like something is going to happen.. Or a cheating in the making. Quite messed up!
Definitely come clean, tell him the truth about looking at his phone. Honesty is the best policy afterall. One little lie, usually turns into bigger lies and more lies.. Lying just doesnt get you anywhere. Let him fess up once you have asked him about it. Yes, hes probably going to be upset you snooped, its a violation of privacy, but you have a right to be upset too. This really needs to be discussed.
Just bring it up as saying you are feeling a little uneasy about the relationhip between them both, dont come off as nagging or attacking him, this will probably make him angry which you dont need after you snooped his phone. lol Just explain calmly you are wondering if they are hanging out or planning to at least. See what he says, if he lies just say well i saw the texts on your phone etc.. See where the discussion takes you and come back here for more advice. I think this sight is really good from an outsiders opinion. As they are not blinded by their love for the particular person and looking from an outsiders point of view is very helpful to see the real situation.
Best of luck!
You have all the weapons you need... Now fight!