First off, something this big should be handled in the group's social contract. At what point is PVP allowed?
An example from my experience:
I played a very blunt human in a D&D campaign and butted heads with an elven Bladesinger. Basically the party was mostly elves and when one would do something foolish, my character would make a joke at that character's expense. Of course, the bladesinger got increasingly offended at this. It kept escalating until swords came out.
First, talk it out
The first thing that should happen is to let the two players/characters talk out their differences. This is likely to be a long process. I suspect your characters have long since passed this point.
Next, let the argument happen
Once talking fails to persuade, someone's going to start yelling. First off, as GM you can cool the argument a bit by letting a wandering monster hear the ruckus and come to investigate. Especially if the "wandering monster" is a group of bandits. People yelling and arguing are likely to not hear them sneaking up for the PC's valuables (and this gem sounds like something that will intrigue many bandits). This is a delicate balance for the GM, you want the PCs to argue and express themselves but if the rest of the group looks bored it may be time for something to happen. Don't immunize the players from the consequences of their own actions, even if it is yelling. After either a fight, or the bandits get away with the loot and the PCs track it down (and fight for it); hopefully cooler heads can prevail regarding what to do with the gem. If not:
I am not sure what is in your group's stated or implied social contract. However, PvP (player v. player) is not allowed at all. Period. However, PC v PC can be allowed as long as both players understand that it is the two characters brawling. My group is a little laid back on using both player name and character name at the table. However PCvPC is an exception to this rule. Only refer to character names while the fight is going on, this helps to reinforce that it is the characters fighting and the players are acting in a dramatic scene.
Once the fight ends, the first thing both players need do is to shake hands, then one or both should leave the table for a few minutes. A bathroom/snack break is a great thing to do at this point. If both players are pros at this, they realize it was CvC and not PvP, if there was a bit of frustration (especially for the loser of the fight), this break lets the player cool down a bit and regain balance.
In my case, after we fought we both got winded/knocked down and were laying next to each other on the ground, and I then said in character, "I was not criticizing all of Elvendom, nor disrespecting [a third character]. I was commenting that a noble character was doing a foolish action." The other player's character then told me why he took offense was because "What a moron" was interpreted as disrespecting the entire race and the character specifically. As we got up, we recovered our gear and made a few snippy "parting shot" style comments that clearly was not nearly as heated as before the fight.
Your two characters may have a series of conflicts over this gem, or maybe one fight which proves to the both that both are resolved in their opinion. And that is a starting point to some great dramatic discussions and roleplaying, IMHO.
However, if your players are not comfortable with PCvPC and it is established in the social contract, then let matters go as far as they can, and when hand touches sword, DM you need to yell "BREAK" or "CUT" or something to stop everything right there. Player's tempers may flare in an argument, but the DM is also the referee in terms of not only the game's rules, but also the group's social contract.