Let me start with a similar statement:
"Another PC came up to me, and killed my character in one shot. I didn't like it, so I said he missed."
In many ways, that above statement is identical in kind, if not in specific consequences to the one you proposed. The fundamental resolution mechanic is that the defending player in this or in the social setting has no way of demonstrating agency over her character. This removal of control is either explicitly or implicitly fought because it goes against the patterns established by the other methods of conflict resolution in the game.
Vincent states:
As far as I'm concerned, the purpose of an rpg's rules is to create the unwelcome and the unwanted in the game's fiction. The reason to play by rules is because you want the unwelcome and the unwanted - you want things that no vigorous creative agreement would ever create. And it's not that you want one person's wanted, welcome vision to win out over another's - that's weak sauce. (*) No, what you want are outcomes that upset every single person at the table. You want things that if you hadn't agreed to abide by the rules' results, you would reject.
This scenario fails due to the latter component, there is no compromise or mutual partial success, as there would be in a combat between the two characters.
If we consider the area of cRPGs, we get Player Skill v. Character skill articulated in an interesting way here:
The mix of player skill and character skill is a defining element of an RPG. In an RPG, you are playing one or more characters – characters who have defined (but dynamic) abilities and limitations that change over time as you make progress through the game.
...
On the flip side, a game can’t be all about character skill either, or it’s not a game. It’s Progress Quest. ... Sometimes you will get rants by inexperienced gamers who argue that there is no player skill involved in in turn-based RPGs. I don’t know if they are just trolling or truly ignorant, but they do say this. I can only assume said kids have never played a game of Chess, let alone completed the original Pool of Radiance.
Now, while he's talking about cRPGs, there is an interesting conclusion to be drawn about typical character-conflict-resolution mechanics. Most games social contracts have a "Don't roll dice against other players" provision, allowing for player persuasion against player. By allowing dice rolls, without the adjudicating components of the NPC game world informed by the GM, there is a one-hit win/loss and the game really does become a sort of progress quest for that interaction. All agency is removed and the losing party feels powerless.
There are three games with absolutely fascinating player conflict resolution mechanics. DitV, Mouseguard, and Ars Magica.
DitV articulates all conflict in one conflict-system, from talking to gun-fightin'. Therefore, if players want to engage in what is PvP conflict, there are ways to interact that provide both sides agency in getting what they want (as well as the ability to roll fun handfuls of dice.) One bad roll won't make someone feel powerless, and both sides can impact the others' reality (do damage) before one ultimately triumphs. Furthermore, there is a mix of player and character skill, as the conflict is rendered both mechanically and narratively. Both sides buy into the rules because they realize that the rules will fairly impose unwelcome outcomes on both sides.
Mouseguard has a similar philosophy, providing an argument subsystem and raising player ideological conflict to a fairly important role. Therefore, players playing mouseguard realize what will happen, and buy into the rules making unwelcome situations for them. Here, there are explicit elements of compromise baked into the system and both player and character skill come into play.
Once again contrast this with kids playing "cops and robbers" wherein one says "bang bang, I shoot you" and the other says "no you didn't." Without mutual buy-in on the rules, this is what a RPG devolves to. Very few people will willingly choose to buy into a lack of agency on their behalf.
Both of the above games can be characterized as narrativistic. However, Ars Magica (a decidedly non-narrative game, depending on how a group plays it) also has a mechanic. Certamen, a non-lethal fight of magic, is designed to provide magi within the game a way to compel action by other magi. It is a fascinating tool for PCs to compel action or inaction by other PCs. While the concept of a magical duel isn't normally perceived as a social skill, it fits the exact same mechanical niche: a way to compel action with mutual buy-in from all involved. This rule subsystem is also extended to debates, offering a way to have players and characters offer points and counterpoints.
Looking at D&D, unfortunately, the lavish attention they put into the tactical combat system is not reciprocated within skills. While skills are a useful shortcut for impacting the GM's world, the abstractions they use do not provide any provision for player response. Some groups I've played in solved this problem by refusing the rules' right to adjudicate inter-player conflicts, and others turned it into a simple attack: skill versus relevant defense.
Neither solution is particularly satisfying. By refusing the rules, players functionally refuse the interesting aspects of entering into unwelcome situations. By resolving the compulsion as attack, there is no symmetry of unwelcome possibility, nor any way of using player skill to impact the outcome.
This, ultimately, is something that must be decided by your group. However, if the group likes the idea of inter-player conflict, it is probably worthwhile importing expanded and useful rules from another RPG to satisfy the dramatic/situational needs of the table.
It sounds like, in this specific instance, it's worth performing a debrief after the end of the game and assessing what the player desires are. Assuming that this was a big deal to both players, I would make up a simple "social combat" system on the spot, importing the tactical richness of the dominant conflict system to this important PvP conflict.
