This is a good question. The United States is important in the world economy in several ways. Let me list a few.
First, the United States is still the largest economy in the world. For this reason, the world still depends on the United States to consume its products. Without this consumption, the world economy would suffer greatly.
Second, America leads the world in the finance sector. This is no small matter. Most of the world still is investing in our markets.
Third, nations are still buying United States debt. Part of the reason for this is because the United States is still one of the safer countries. Think of the EU and the problems its faces due to Greek, Italy and Portugal. You can say that there is a flight to safety.
Finally, the dollar is still the world's currency and equally as important oil is priced in dollars.
In short, the United States is the single most important economy in the world. In the past the United States lead growth. In the future, I will lead with China.
The USA is definately still one of the top countries in the world in terms of economy, but no longer the best. In fact, its power and its influence to the world is actually subsiding slowly(compared to other parts of the country). It has became a dangerous factor. Since it owes the other countries so much, if its economy was obliterated, other countries, especially China, that lent money to the USA would be greatly effected.
It is undeniable, however, that it's still one of the most powerful countries in the world, but the US economic system has already determined the ineviable fail of it.