They reacted carefully and cautiously, but poorly and ineffectively. While the Special Relationship had existed for a long time, so had the Monroe Doctrine, of which I've spoken about here, and so had the Organization of American States (OAS).
Don't get me wrong, US foreign policy has only ever cared about the Monroe Doctrine when it suited their interests, turning a blind eye or outright supporting European colonial attempts when the outcome was considered to be potentially favorable for the US government. But in the context of the period in which the Malvinas war exploded, the situation was tricky.
You see, since the late 40s the United States military had been training Latin American military officers in guerrilla warfare through the School of the Americas with one very clear purpose: to establish different military dictatorships that would serve as protection against the dreaded possible advent of communist regimes, a fear that only became worse following the Cuban Revolution of 1959. One LatAm country that evidently followed this very peculiar set of guidelines that we've come to call the United States' National Security Doctrine was Argentina, under the de facto rule of the last military dictatorship that seized power by deposing president María Estela Martínez de Perón in 1976. Their justification for taking over the country and eliminating the rule of law was the fight against insurgent terrorist groups that were acting in certain regions of the country. Thing is, by 1977 said guerrilla groups had effectively been exterminated. Still, the military continued in power, making people think that there was still a terrorist threat, that the livelihood of proper, obedient citizens was still under imminent threat. Which was, of course, bullshit.
When that started to fail, when people started questioning the reasons for their continued de facto government, the Junta had to come up with other excuses, all of them being what we call nationalistic callings, propaganda campaigns that continued to pit the general population against a fabricated enemy. The one most widely use was still the threat of subversive terrorists living in your neighborhood, or being manipulated by marxist teachers at your kid's school.
However, there were others. During the 1978 World Cup, the enemy were the Netherlands team and the foreign press, who were, according to the dictatorship, colluding with the enemy. Why? Because they had decided to talk to the Mothers of Plaza de Mayo, a human rights advocacy group of mothers of the disappeared, the tens of thousand of people who had been kidnapped, tortured and executed by the military on suspicion of subversive behavior. After that, there was the territorial conflict with Chile over the Beagle canal, which was also bullshit and an excuse for two dictatorships to have a military-measuring contest. And after that, came the Malvinas.
You see, by 1982 the military had already lost almost all the support and legitimacy they had managed to build in the first year of their rule. People were tired of their excuses, more and more foreign denunciations kept piling up, more and more people kept disappearing, and the recession, caused by the implementation of neoliberal policies that deeply endebted the country to foreign creditors was making life increasingly difficult for the vast majority of the population. And so, Malvinas became the cause behind which the military tried to rally the people, one desperate last ditch effort to create a patriotic feeling among the people that would, in the mind of de facto president Leopoldo Fortunato Galtieri, guarantee a renewed support for the just and holy cause of the military dictatorship. In a way, Malvinas was as much a last attempt by the Junta to desperately cling onto their fading power as it was for Thatcher.
Why is all of this relevant? Well, because the US needed Argentina to remain a bulwark against communism, but at what cost? Was it still worth it? Was a barely surviving dictatorship worth the trouble of getting into trouble with the UK and therefore with the rest of NATO, or was it more important to let the dictatorship fall and concentrate in other, more stable and "legitimate" dictatorships like Pinochet's in Chile?
The answer is, they weren't sure. That's why I said carefully and cautiously. The US didn't want to fight Britain, but, at least at first, didn't want to aid the dictatorship too much either. Their compromise was to support the UN Security Council's Resolution 502, demanding the immediate cessation of hostilities and the immediate withdrawal of Argeninian forces from the islands, in the hopes of being able to act as mediators between the two countries.
And yet, what if diplomacy should fail? Because, considering that both countries had engaged in open hostilities, diplomacy failing was certainly a likely outcome. There were two sides in the US, one of them considered that the US would have to support Britain. This sided included Reagan, Caspar Weinberger, the US secretary of defense, and Alexander Haig, the secretary of state. They thought that the moral and territorial claim issued by the Junta was not enough to justify keeping the islands, but most of all, they were terrified of the implications not supporting the UK could bring for their trade and military ties. On the other camp, we find people like Thomas Enders, the undersecretary for Latin American affairs, and Jeane Kirkpatrick, the US ambassador to the UN. Did these men think that the US should support the Argentinian dictatorship? Absolutely not: they thought that the US should remain neutral, because neutrality was key to maintain the balance in the fight against the possible Soviet advances into both Europe and Latin America. You see, they thought that if the US were to openly support Britain, the other Latin American dictatorships would lose all confidence in the US' commitment to the Monroe Doctrine and the defense of the continent against colonial incursions. Their thinking was that, since the geographical area in which the conflict was developing released the US from any of the obligations NATO would've otherwise imposed, they were safe.
The result was diplomatic chaos. The US officially endorsed a diplomatic solution, while being entirely incapable of hiding the fact that even while they were allegedly acting as mediators, they were supporting 10 Downing Street with supplies and intelligence. As a mediator, Haig was, let's just say, bad. He failed to hide his pretty obvious support towards Britain in his visits to Buenos Aires, and never properly communicated the demands from each side in an ubiased way, leading the Junta to realize, altogether too late, that the US government was merely trying to buy the British time to organize their counterattack.
The US' behavior was one of the factors that led to the Junta losing the war, and therefore eventually losing power in 1983, but it had deep effects in their relationship with London as well. While Thatcher and her successors were and are very much aware of the support the US provided during the war, the British public wasn't too keen on what they saw either as faux neutrality or as outright treason. Peter Calvert wrote a pretty interesting article in 1983 about that called Latin America and the United States during and after the Falklands Crisis. However, for more on the whole topic I strongly recommend David Welch's 1997 article Remember the Falklands? Missed lessons of a misunderstood war.