So to start with I don't like Paladin PCs. Did not like them in 1Eor 3E, 5E is a little better but they still have that silly oath. They are great for that knucklehead NPC.
The oath is just fine if you want to play good guys - the problem here (as many have stated) the CN players in your group are not CN they are evil - full stop.
We have an oath of devotion Paladin in a game I am DM ing. His Deity is Tyr. He is Lawful good, the rest of the party is Chaotic (CN fighter, CG warlock/rogue and CN Barbarian). As an aside I am not a fan of CN players either but that is another topic.
The problem is that the charactets wrote CN on the character sheet but are playing CE. From the described situation THEY'RE fueling the conflict not the paladin - who (again, from what's written) is playing his oath and his alignment properly.
Well there is constantly friction in the party as to how to handle things. For example they typically want to kill prisoners after interogating and promising to let them go, they have no problem lying and well acting Chaotic.
That's evil not chaotic. Of course there will be conflict if a party member wants to prevent evil and the others want to instigate it.
The Paladin tries to role play his character with his oath but it is causing a lot of friction. It came to a head in the last game when they let a stone Giant go after they got done arguing with him. The two CN characters wanted to kill him because they thought he would come back and attack them. the Paladin stood his ground and said no way and drew a line in the sand, the CG warlock rogue sided with the Palidin but would have been fine with "looking the other way" if the Paladin did not make a big deal out of it. OF course they were ambushed by said giant and friends later that session. It could have been a TPK, and would have but I played the Giants poorly in battle, making some bad decisions and the party scraped out of it (barely).
The evil characters are causing the friction, not the paladin
I am really having trouble with this because I think in real life those characters would just part ways - the Paladin can't accept such behavior and the others can't stand the goody-goody Paladin. Of course in the real world parting ways means an end to our game. If it wasn't for his oath I think he would just relent and basically look at alignment as a guideline or belief instead of a code to live by.
I agree, the characters presented shouldn't adventure together.
BUT
It's not the oath, any good character would (well should) have a problem with the described evil actions. Trying to state the problem with the paladin player is misplaced.
As for solutions :
Make sure you and the players are on the same page. Personally, I think the best way is for the players to recognize evil when they do it and if evil is not the intent - then change that behavior.
But if you and they are OK with the actions - maybe have the paladin change to oath of vengeance or even conquest. Those would work much better with this group.