Hi. There's nothing wrong with taking your time to watch the situation. It's your marriage, it's important, you don't have to rush.
If I were you, I would be wary of letting it simply fade away though. Those "why" questions were never answered (why does he seem so strongly attached to her, why is he glued to his phone, etc.). If something is going on between them, or starting to go on, your DH may think that he just needs to keep it a bit quieter to avoid problems with you.
Especially if he and she feel like this is the very beginning of something that may be really special and they really have to find out what it is, he may get very protective of it. He may start to feel, however ridiculous this sounds, that he owes it to himself to find out what this thing is between them, he may therefore decide it's okay for him to keep it going very quietly so he can find out "what it means." I hope that is not what's happening, but mention it so that you'll be careful not to let this problem fade away. I know it would be nicer to just let it go but I think that is risky.
So again, don't stop watching, and do sit him down again when you're ready. At the bare minimum, meaning even if you find out for sure that there is/was nothing going on between them, you'll need to talk together about boundaries, what's appropriate in a friendship, making each other the priority, etc. Good luck.