I have to be honest, I've never got this whole "my phone is private" business.
I wouldn't have an issue with someone looking through my phone and I think, if you have, then you've got something to hide.
You're his wife, For Funks Sake, you can nose at whatever you like. If he gets defensive and cagey, then clearly he's hiding something.
Saying he wants to be in her warm bed would be the tipping point for me. I'd shove that phone where the sun doesn't shine!
Realistically speaking though, you are married and it would be a bit daft to end the marriage over something as silly as this - that's assuming you are correct and he *hasn't* met up with her!
Ok, so you know she's on his Facebook list. If it were me, I would send her a "friendly message". Tell her you've seen the texts they've been exchanging and that if it doesn't stop you will have no hesitation in informing her husband.
Oh, and add the husband to your friends list too! That should stop her.