For information or intelligence in Middle East the US can get not just Israel but Saudi too(less reliable but still option). Also Mossad is clever and “ears on ground” directly. There intelligence is focused there and have direct interest in geopolitics.Essentially, as people have already said, this is unlikely to the point of ASB.
However, let's accept that ASBs somehow have USA decide to back Argentina in the conflict to a similar extent that they backed the UK in OTL. That is, intelligence and moral support, without actually getting involved.
Since intelligence is being shared between Argentina and the USA, the British Mission in Washington promptly shuts down. This was (and is) by far the closest intelligence cooperation between two countries around, with information from assets being shared. This pretty much blinds the USA in the Middle East (where the vast majority of assets were British) and the Indian subcontinent (where the USA was pretty much totally dependent on Britain). Building a network is a lengthy process, and American intelligence would be pretty much Israeli-dependent for the next decade and a half in the Middle East. Good luck with that.
NATO undergoes strains. America is backing a non-NATO country in a war with a NATO country. That is going to send shock waves throughout the whole of the organisation. If America decides to support attacks on allied nations within NATO, you can expect a lot of problems. It might well lead to a development of a European force. NATO had already been under strain over Cyprus; this would be an order of magnitude greater.
US bases in Britain were already controversial. We're in the run-up to the height of the Greenham Common protests already. If the USA has backed an enemy of Britain in a shooting war, expect public pressure for these US bases to be kicked out to mean that any party that didn't back that to fail in the polls. An election is going to happen by 1984 at the latest, and the only question would be whether the US servicemen would be home for Christmas.
With France, Ireland, and now Britain pretty much out of NATO at various levels, the strategic situation in the Atlantic shifts. Spain isn't yet in NATO, and joining wasn't popular there. While Spain has sympathies with the Argentine claim, you can bet your bottom dollar that the evidence of America joining the fight against a NATO country (as it would be portrayed) would be politically impossible. That's pretty much the entire west European coast now no longer friendly disposed towards the UK.
After a lengthy bout of France saying "I told you so" to Britain, it's probable that close military cooperation between France and the UK outside of NATO would start to develop.
The northern flank of NATO becomes a mess. Defence of the Norwegian north was pretty much the purview of Norway and Britain (I should know. I invaded the Soviet Union from there). That's likely to fall apart.
On the other hand, the USA gains the support of Argentina. Which was, at the time, an economic basket-case with a modest military capability in a strategic backwater. Still, at least Argentine can continue with its Dirty War without being disturbed.
As for the course of the war, not a lot will be different. The US no longer supplies Britain with the latest model of Sidewinder, but since air control was never contested, that's simply going to mean that a few Argentine pilots make it home after disengaging from aerial combat rather than getting splashed. If the US were to supply Argentina from NATO stocks (as it did for Britain), all hell would break loose. WE can discount that. American advice would have been listened to more carefully, which benefits Britain, as British and American operational tactics were very different; Argentina would prepare even more for the wrong approach, and have an even harder time adjusting to the actual situation on the ground.
But it's silly, because the original concept is silly. Despite Kirkpatrick (more fool than knave), there was no chance of America turning on Britain.
Given “intelligence” UK provided US in places like Iran there lost of intelligence is not huge lost. Israel and Mossad would be happy those.
Also US has military bases across Europe especially in Germany which during Cold War was important but honestly resource money drain after. Even during Cold War the only benefit of bases in UK is navy bases and more areas for Soviets to send their missiles before they even think of hitting our actual land. Also not smart to concentrate all or too much of your forces to one place in case of missiles exchanging. Other then that it’s just power projection and excuse/bailout for Europeans to give more money to their own welfare state instead of military.
The US would not send anything that would start open conflict with Brits. Just cut money flow and harm a aggressive PR campaign against Thatcher and her supporters to point of pressuring them to pull out. Like French in Algeria Brits could militarily but still lose politically.
It is good to remember that Americans got Brits to sign over all their North American bases for lend lease. They got outdated boats and 90 percent discount on goods but US ended their empire in Western Hemisphere without a single shot. This is why uk has to often be clever and bait US into stuff because otherwise their public truly does not care or even agree or think like them half the time. It’s more so just old money and elites that exchange between these two especially later in century we get