It would be political suicide for any British government to capitulate so they won't. And it would be foolish and tragic for any Argentinian government to offer up its soldiers to the slaughter for the sake of a useful domestic diversion.
I'm not sure how, exactly, the Royal Navy is going to slaughter anyone. They simply don't have the units needed to cross the Atlantic, dominate a contested area and land the Marines. You can't do what you can't do, no matter how daring your aviators, how bold your sailors, how brave your soldiers.
It's not going to come that because Argentina does not have the military it would need to overwhelm [1] even the sparse garrison (two companies of infantry, a ship, four fighter planes) the Brits maintain in the Falklands.
[1] Of course, surprise happens in war, all the time. Argentina could pull a Pearl Harbor on the Falklands and then where would the Royal Navy be?
