I swear I've met squirrels with bigger brains!!
It's not that, at all. He will completely understand why it's inappropriate, but you can't argue with stupid, so he'll pretend that he can't understand it no matter what.
Then you back off because he doesn't understand, and you love him, and he's a good dad, and you don't want him to kick off (delete as applicable), and he gets to carry on having what is, at the very least, a presumptuous and very flirty conversation.
I think you're likely to be on borrowed time, to be honest. Their flirting is very real, lots of open compliments and mutual attraction. Even if that's all it is, at the moment, they've both made it clear that they'd be interested, and they go drinking a lot which tends to lower inhibitions. He is playing with danger going drinking with someone that he's attracted too.
"Stocking filler" is almost irrelevant, as a contact name. It's gross and it shows his view of women, but it says nothing other than that she is special enough to him to have a special petname, something that makes him smile when she texts him. That's the bit that I'd worry about.
I would ask him who Stocking Filler is, and then wait. I know you know, but if you let him start talking and stay silent, he'll probably fill the gap..."Oh it's just X....pause....it's her nickname/it's a joke/whatever rubbish he's likely to spout...."
If he is happy for an uncomfortable silence, or if he turns this on you for snooping or says he knew you'd get upset or any of the other minimising bollocks, you know that he doesn't care about you feeling uncomfortable and unhappy, and that's not good.
Hopefully the conversation will tell you what has been going on, and if nothing terrible has happened yet, it'll show him how wrong this is. He needs to realise that himself though.