The US did back Britain, The US declared support for the UK and sanctions against Argentina on April 30th 1982. Due to the nature and location(Argentina not being attacked, UK not being attacked in Europe or North America) of the conflict the US was under no obligation to support either side militarily despite being allied with both.

The US was required by agreement to allow use of some US military bases by the UK during the Falkland war

https://history.state.gov/milestones/1937-1945/lend-lease

On September 2, 1940, President Roosevelt signed a “Destroyers for Bases” agreement. Under the terms of the agreement, the United States gave the British more than 50 obsolete destroyers, in exchange for 99-year leases to territory in Newfoundland and the Caribbean, which would be used as U.S. air and naval bases.