Legally - that is state to state, I think. Morally...? Guess only you can answer that.
I can tell you this: He is doing nothing wrong! It is not illegal for him to break his vow. It is not illegal for him to cheat. It is not illegal for him to give you an STD or to have a child with someone else while married to you. And in most states none of the above is considered immoral either.
The courts in most states now) therefore, offers a BS no protection, nor any compensation should any of the above occur - it does not punish the WS in anyway. Marriage is not protected as it once was.
Example: My WH and I both signed an agreement after Dday to a division of assets and support - should we D. This was his way of assuring me I would not be screwed over as I considered R. This contract as it turns out had no legal standing.
Now when a contract is broken by say - a roofing contractor - the homeowner has recourse in court. Not so with a contract between spouses! Which still amazes me because the vow of "forsaking all others" -- was a verbal contract with how many witnesses...? 100 or more? It doesn't matter - The court does not care.
Now you can follow him, hire a PI, sit outside his office and listen with a glass up to the door... Nothing illegal there. Immoral...? Again only you can determine that. Will those methods work... maybe.
Question: If you found out that you H was afraid you were cheating and lying about it, so he was recording your conversations - would you consider it immoral? I wouldn't have. I'd have found it weird and a tad creepy perhaps, but not immoral. I'd be more concerned that he was that worried about my fidelity.
These are my thoughts on it.