But on the other side of the coin. With Britain being a member of the European Union, wouldn't a "first strike" by Argentina on the Falklands trigger the EU treaty and require the entire European Union to participate on the side of Britain? Come to think of it, this entire thing could escalate into another World War...
That is a possibility. During the event of which Obama would no doubt side with Argentina, because that would be the Obama thing to do
The European Union is not a military alliance. An attack by Argentina would not trigger a European response.
Nor would it trigger a NATO Response as such responses are limited to attacks on particular geographical areas that do not include the Falklands - it is the North Atlantic Treaty.