Simply not true I'm afraid. The Americans appeared to behave very coolly towards us when we announced our intention to retake the Falklands by force - witness US Secretary of State Alexander Haig's seemingly increasingly desperate 'shuttle diplomacy' in the run up to the war - and their opinion that the war was 'unwinnable' but the reality was very different.
Firstly, while Ascension Island is technically a British Overseas Territory, it had been leased to the US as a strategic base; both for monitoring satellite communications and general SIGINT and also as a handy runway for their (brand new) Space Shuttle to use in a emergency. The Americans agreed to let us use it for our Vulcan bombers, Victor tankers, Nimrod maritime patrol aircraft and Hercules transport aircraft. The island's location was also a very useful stop-off point and staging post for shipping and supplies on the way down to the Falklands.
Secondly, not only did they let us use the airbase on Ascension, they provided a constant supply of vast quantities of aviation fuel to our aircraft there. When the British commander told the US military commander on Ascension how much fuel we'd need the US commander remarked that we'd never use that amount; the British commander replied that it was certainly our intention to try! Don't forget that it took 11 Victor tankers to get the Vulcan in position to bomb the runway at Port Stanley!
Thirdly, the US made available the latest, most technically advanced version of the AIM-9 Sidewinder missile; Specifically the AIM-9L 'All Aspect' version. Previous versions of the Sidewinder needed to be fired towards the rear of the target aircraft - the hot jet exhaust - in order to have a decent chance of scoring a hit. Having to get on the tail of a enemy aircraft in a dogfight rather than just getting it on one's sights at any angle adds greatly to the difficulty and danger of a dogfight. This new 'All Aspect' version could be fired at a enemy jet from any angle (hence the nickname!), greatly reduced this risk and provided the Sea Harriers with a significant advantage. That is not to say that the missile did all the work, but it did provide a massive boost in capability.
Fourthly, the US also made available to us the Shrike anti-radiation (radar) missile. We had been investigating using the Martel missile attached to the Vulcan to attack Argentine radar installations on the Falklands but the Shrike was a much more capable missile. Several of the later Black Buck Vulcan raids (including the infamous one where a Vulcan had to make a emergency landing in Rio due to its' inflight refueling probe breaking off) were launched with this purpose, taking out several Argentine radar installations on the island. This enabled the Sea Harriers (and RAF GR3s which joined later on in the conflict) to operate with more impunity in the air above the Falklands and provide Close Air Support (CAS) to our forces on the ground.
In short, the US kept up appearances that they were not in support of our Falklands campaign, but behind the scenes they did provide some significant assistance which probably proved vital to the successful outcome.