I actually think from your OP that now is NOT the time to address this. You need to calm down first.
If you've never told him that you'll be checking then he WILL feel like you have invaded his privacy. So if your intent going forward is to continue to check (and address what you find) I caution you that telling him you're going to check may only lead to sneaky behaviour (deleting, private browsers etc).
As others have said, the way he speaks to you and how he speaks to his friends is bound to be different, and that's usually okay. Usually more swear words, a lot of bravado etc etc
If you ARE going to address it, I would do so calmly (your OP does not come across that way at all). I would talk about how words can hurt and that you assume he never meant for this girl to hear or see those words but sometimes we don't have control over what people see or hear, no matter whether we think we do or not, so it's best to be kind whenever possible.
I would also address the lie the girl told and tell him that it's not okay that she's lying about him, that you understand why that upset him but that he also doesn't know whether his "friend" (or someone else) is lying about what the girl said (she could be a victim of the lie as well) and other people could say he's said stuff when he hasn't. So treating her poorly isn't okay. I would also discuss ways to deal with it if it turns out she did lie (ignore, deny, laugh).
I would also ask him how he would feel if it turns out she did NOT say she had kissed him, and then his friend showed her those messages. How sad that would be for her for someone to say something nasty about her for no reason (he was upset that "she said" something about him for no reason). She could be extremely hurt and upset either way that that's how he thinks of her and he could get into some trouble at school etc if his friend shares the messages around.
Perhaps also a discussion about how a real friend wouldn't screenshot that message and send it to someone else to get him into trouble so it might be time to spend some time away from this particular boy.