in fact, there has been a lingering conflict between turkey and greece, both early nato members, since before they became nato states, and it became a hot proxy war during the invasion of cyprus, which is actually a fairy close parallel, as cyprus was a greek-supported semi-autonomous region at the time.
turkey has a much larger military force than greece, but the greeks are notoriously hard to conquer. the turks failed to conquer cyprus in the 1970s.
it's not clear how denmark could provide any sort of resistance to the united states. while the conflict should provide a framework on how to deal with conflict between member states, it's also not likely that a un peacekeeping force would help much, or do much besides provide an excuse and cover for the subsequent canadian invasion and annexation of greenland.