The Argentines have probably noticed that the Brits no longer have the long-range Vulcan bombers that hit the islands in the Black Buck raids or the Victor tankers that refueled them on the way. Similarly, all but one of the HMS Invincible-class STOVL carriers in the Royal Navy have been mothballed or scrapped, and the remaining one has been de-clawed, since the Fleet Air Arm is utterly bereft of the stars of what may soon be referred to as the "First Falklands War", Sea Harriers.
To put it bluntly, the British no longer have the ability to project power across a hostile shore over transoceanic distances.
And does anybody think that Obama would back Cameron over the issue the way Reagan did Thatcher? Yeah, me neither. As a matter of fact, a shooting war on British soil, (even if it is British colonial soil,) to which the US responds tepidly to its treaty obligations would make NATO look pretty toothless.