Who would win is irrelevant as everyone would lose;
But how would other countries react? Which side would NATO take?
It's just a slow friday afternoon in the office
The best case scenario is that neither party invokes its rights under Article 5 of the North Atlantic Treaty (see https://en.wikipedia.org/wiki/North_Atlantic_Treaty#Article_...), which is the bit that requires signatories to treat an attack on one as an attack on all. That doesn't happen automatically; NATO would only be obligated to act if one of the fighting parties called on it to do so by invoking their Article 5 rights. If neither of them do that, the rest of NATO can just avoid the awkward question altogether and let them fight it out.
If one of the fighting parties invoke Article 5, then NATO is in a bit of a bind. At that point they can't ignore the issue anymore, they'll be obligated to either line up behind the signatory or be in breach of the treaty, rendering it null and void. My guess is that, unless the aggression is seriously noxious, the most likely result is that NATO would just ignore it anyway and hope nobody cares too much. If the country that invoked Article 5 doesn't have many friends in NATO, it's possible that the other countries could just pull a discreet veil of silence over their failure to live up to their treaty responsibilities. A more likely scenario is that the failure results in one or more signatories pulling out of the treaty, and NATO breaks up.
If both of the fighting parties invoke Article 5, then we're well and truly in uncharted territory. The North Atlantic Treaty doesn't specify that it only applies to attacks on NATO members by non-NATO members. Again, the most likely scenario here is that the other members decide not to uphold their treaty obligations and NATO ceases to exist.
Note that the odds of any of this actually happening are smaller than they might seem. Beyond the obvious (i.e. the members of NATO are all pretty friendly to each other, with few casus belli hanging out there), there's also the fact that Article 6 specifies that the collective action responsibilities only apply to attacks
> on the territory of any of the Parties in Europe or North America, on the Algerian Departments of France 2, on the territory of Turkey or on the Islands under the jurisdiction of any of the Parties in the North Atlantic area north of the Tropic of Cancer;
> on the forces, vessels, or aircraft of any of the Parties, when in or over these territories or any other area in Europe in which occupation forces of any of the Parties were stationed on the date when the Treaty entered into force or the Mediterranean Sea or the North Atlantic area north of the Tropic of Cancer.
So if two NATO members get into a scrap over some old colonial claim in Africa, say, neither one could call the other treaty signatories into it. It's explicitly limited to just attacks that take place within a pretty narrowly defined geographical scope.
(A scope that's sufficiently narrow to raise some interesting questions. For instance, it arguably would not include an attack on the United States if that attack only struck Hawaii: https://www.westernjournal.com/nato-loophole-attack-hawaii/)
