Since 1982, it's been a piece of conventional media/political wisdom that Britain prevailed in the Falklands War because of invaluable American support. This is especially true of Margaret Thatcher, both in office and in the post-ministerial Thatcher industry, as well as a wide range of pundits, Tories, and others. Up to a point, it's also true that a lot of people on the Left seem to believe that "US support" somehow caused the war, operating on the common heuristic that Europeanism is a sufficient condition of pacifism.
For the frothing Right, of course, it's a truism that we needed US support because the Europeans "betrayed" us - some people will even claim that France sent more weapons to Argentina during the war, which is (as we shall see) wildly counterfactual. Both groups use their specific versions of the myth to justify their general policy prescriptions.
So how great was that US support? Politically, it wavered throughout the war. The US government was understandably unkeen to see their pet dictator and closest military ally get in a fight, and some people (notably Jeanne Kirkpatrick) seem to have thought that a British fiasco would not have significantly damaged the alliance, while a triumph for the Argentine junta would have dramatically strengthened their hold on power. Perhaps a disaster for the British would, indeed, reinforce their dependence on the US. Hence, the net balance of US interests lay on Argentina's side.
However, this view never carried the day, and it seems to have been stronger at the State Department (and presumably the CIA) than the Defense Department, especially the uniformed military. The US moderated its position somewhat as the first British forces reached the South Atlantic, and stayed that way.
Materially, it's traditional to thank the US for the use of the Ascension Island airfield. It was indeed vital, but after all, it was our airfield, and the RAF began using it without waiting for political approval. The total US presence consisted of about a dozen men, mostly civilian employees of PanAm. The RAF had to bring all its own logistics, including getting the Royal Engineers to build a pipeline from the tanker landing point to the airfield. Speaking of which, the US also supplied a lot of jet fuel under the existing fuel-exchange agreement. Anything less would have been astonishing (they didn't, for example, cut off the fuel exchange deal at Suez as far as I know), and anyway jet fuel is available in commerce.
There was also some kit, specifically the newer AIM-9L version of the Sidewinder air-to-air missile which was already on order but was delivered quicker. Again, it would have been a serious breach in the relationship to refuse this, and anyway, the Fleet Air Arm was ready to go to war with the missiles it had.
It was traditional to refer, if pressed, to the intelligence special relationship, the most secret and elite heart of post-war Anglo-American relations. Here are all those sets of initials - UKUSA, CAZAB, ECHELON - denoting treaties whose contents remain concealed from the grubby mob to this day, 59 years after UKUSA's conclusion. Surely there must have been vital secret intelligence?
Here's the shocker. Early in the war, a British delegation containing the then Defence Secretary, John Nott, and the Government Chief Scientific Adviser, Ronald Mason, visited Washington to ask for satellite reconnaissance of the Falklands (and presumably the Argentine southern air bases too). They were refused. It is widely believed that the basic content of the UK-US intelligence agreements includes provisions to the effect that in return for the use of British intelligence product, the US would provide access to things like satellites we don't have. It seems that the argument from spookery was always quite simply an appeal to things mortals are not allowed to know.
Another question hangs over the offer of a US aircraft carrier for the Royal Navy's use. The idea was apparently that the RN would crew the ship and provide its air wing. This is often cited as an example of US generosity, but this is naive. The crew requirement alone would have be huge, and everyone would have needed to requalify on equipment they had never seen before, not to mention learning to handle the huge unfamiliar ship in company with the fleet. No Royal Navy pilots had carried out arrester landings since the old Ark Royal had been decommissioned five years ago, and all the suitable aircraft (F4G Phantoms and Buccaneers) had been transferred to RAF squadrons and stripped of their specialist gear. What of the secret equipment? Crypto? Refuelling at sea? Given that the whole operation was governed by the need to finish the job before Admiral Winter got involved, it is hard to avoid the conclusion that this offer was a white elephant in the original sense of the phrase - a gift intended to exhaust the recipient.
It's a mixed picture, frankly.
What about the Europeans? The then EEC joined in a trade boycott of Argentina, even if the whole thing was rather divisive within Europe. France offered extensive intelligence and diplomatic help to round up stocks of the air-launched Exocets that might otherwise have been sold to Argentina, a crucial point. French naval aviation also took part in a discreet project to develop counter-Exocet tactics, using their Super Etendard aircraft to simulate attackers for British ships and planes in the North Sea, and opened the books on all the equipment they had sold to Argentina. The Swiss manufacturers of the Skyguard and Super Fledermaus radars and Oerlikon 35mm guns did likewise. The French Air Force also opened its staging post in Dakar to the RAF.
Conclusions? Well, if the Falklands experience is meant to be a guide for future policy, the obvious lesson is not "The Americans are great - let's rely on them even more." Neither is it "I don't need a defence policy, I'm pro-European." More seriously, it shows that the key characteristic of any policy at all is adaptability - what Rupert Smith calls organisational mobility. It's not going to be like last time.
Further, don't count on "the West," "the integrated core," "the world of order", "the democratic world", or "the Anglosphere" to know what to do if a conflict breaks out within its own walls. Certainly, the US considered Argentina part of the Western system at the time, and you can just about make a case that it used to be in the Anglosphere. But nobody had the faintest clue what to do about it. There was no measurable pressure to resolve the conflict beforehand, nor did anyone (this means you, Uncle Sam) try to restrain the Junta or warn the British. Nor did anyone do anything effective to stop the British "going to war with horse, foot, and guns in the year of Our Lord 1982!" as Max Hastings put it.
Meta-historiographical note: This myth nearly contained another myth. I thought I remember that Belgium refused to sell the British Army more ammunition in 1982. Googling, I learn that this in fact happened in 1991, but the myth seems quite common. It's a pity because I had to lose quite a good joke as a result.