The US did provide military assistance to the UK and none to us, so I'd say they definitely took the UK's side [1].
Complicating things, of course, is the fact the US saw our military junta as a bulwark against communism, and so provided some support to them [2] (and, it is argued, our military dictators thought they saw indications the US would support us in our war against the UK, which is just as deranged as it sounds. Then again, our dictatorship was as incompetent as it was bloodthirsty).
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[1] https://en.wikipedia.org/wiki/National_Reorganization_Proces...
> "However, relations soured after the U.S. supported the United Kingdom in the Falkland Wars."
[2] https://en.wikipedia.org/wiki/National_Reorganization_Proces...
> "The United States provided military assistance to the junta and, at the start of the Dirty War, Secretary of State Henry Kissinger gave them a "green light" to engage in political repression of real or perceived opponents."
> "The US Congress approved a request by the Ford Administration, to grant $50,000,000 in security assistance to the junta. In 1977 and 1978 the United States sold more than $120,000,000 in spare military parts to Argentina, and in 1977 the US Department of Defense granted $700,000 to train 217 Argentine military officers."