Not entirely, in either sense. The Euro problem is an American problem in the sense that either economy can pull the other one down. Much of the fiscal crisis in Europe is tied to the events which lead to the 2007-2008 melt down, which was of American origin.
That is only part of the picture, but it is a part of it.
The world is a small place. That is why I can sit at a computer in Germany at 19:35, and type messages which will appear almost instantly on a server in California at 10:35. We are all linked. More and more, what happens on some remote part of the world, will affect some other part of the planet, and very quickly, in historic terms.
Ask any epidemiologist about this. Epidemics spread worldwide at an astonishing rate, thanks to cheap and plentiful air travel. It is in US and European interests to fight disease in Africa and Asia.
We are all in this together. International borders and geographic boundaries mean less and less as each day passes.