The UK's armed forces could not repeat the heroic recovery of the Falkland Islands in 1982. We had 2 aircraft carriers and a fleet of Sea Harriers then and came perilously close to losing - closer than the press of the day could report. Now we have no carriers or Sea Harriers and are spread too thin to be able to mount another recovery, should Argentina have the appetite to exercise what it claims has always been its property. The USA did not immediately support the UK's efforts in 1982 (as it had just signed trade agreements with the Argentines, I believe), so any support from them would depend on other factors.
The sovereignty of the Falkland Islands has been an issue for such a long time and I think I am right in saying that the islanders themselves want to remain dependent on the UK. Rather than let the discovery of oil create animosity in the region, maybe the resource and its development could be shared in some way with the Argentines provided that the islanders themselves benefit. But maybe the islanders don't want that 'benefit', and if somewhere is worth fighting for then it is worth keeping, let alone what a sell-out would mean to the memory of our forces that fought and died there in 1982. A difficult question indeed.