The most likely short term impact is that the European sovereign debt crisis will cause the Federal Reserve to refrain from raising interest rates. The bloomberg.com link below says that the President of the Federal Reserve Bank of Chicago has indicated that
the European sovereign debt crisis will prompt the U.S. central bank to delay raising interest rates.
The reason for this is that the crisis is likely to depress economic activity worldwide. It does this because it decreases Europe's buying power thus making Europe conduct less business with the rest of the world.
When an economy faces a situation where economic activity is likely to decline, its central bank should not raise interest rates. Raising interest rates makes it harder to borrow money. When less money is borrowed, fewer goods and services are bought. Raising interest is something a central bank should do when there is a threat of inflation, not when there is a threat of recession as there is now because of the European sovereign debt crisis.