The major thing that the European sovereign debt crisis might do to the US is to depress economic activity in the US. This is because the European sovereign debt crisis could depress economic activity worldwide and that impact would be felt in the US as well.
If European countries like Portugal, Greece, and Ireland default on their debts, for example, the whole European economy would slow down. As a Fed governor says in the link below,
A deeper contraction in Europe ...would have the potential to stall the recovery of the entire global economy...
This would mean that there would be less demand from Europe for American goods and services. This would hurt the US economy.
Since Europe is such a major part of the global economy, events that hurt the European economy are likely to hurt the US economy as well.