Wait it out
I would just wait it out. You definitely violated a huge breach of trust yourself by reading his step work. My RAH explicitly asked me to please respect his privacy and not venture near his recovery notebook. And I would never dream of doing so. I have experienced that type of snooping behavior and it can be very damaging. He will feel violated and that will become the primary issue, not the issue of an alleged affair. Maybe he intends to come forward later into his step work? I think that his recovery should be his primary focus right now and working the steps with his sponsor. Give him the time and space to do that while you focus on yourself, working your program and not his. Trust issues do not solve themselves in a day anyway. If you confront him, you would have to be prepared to act on the consequences, leaving him, or staying with him and working through it together. It doesn't sound like this is something he is yet ready to do in his recovery. The horse must come before the cart after all.
Alcoholics and addicts do terrible, awful things under the guise of their disease that they would never do otherwise. This does not make their actions acceptable and does not get them off the hook for the consequences. It just makes things very confusing for their SOs when they decide to pursue recovery! The damage is done. But you also want to move forward. It's a tough call. I wish you the best of luck.