The fundamentals have not changed, yet:
#1- The US and other countries still compete with countries, such as China, that can unfairly exercise downward pressure on wages and their currency. This brings living standards down, reduces wage growth, and leads to job losses. As I have said countless times before: trading with China will lead you downward, them upward, until balance is reached, at which point trade levels between the two will normalize. This is only now starting to take place, as the Chinese economy begins to deteriorate, which will lead to higher levels of spending per capita by the government and boosting internal consumption. Basically this is the start of the Westernization of the Chinese workforce, with all the associated costs. But all of this will have come at a cost for the US over the past many years, and years to come until balance of trade is reached (which should, by then, have reduced the trade deficit).
#2- Students are still massively unemployed and debt-saddled. This will only begin to change as more people retire, but people are retiring later than ever, and the cost of retirement on the US government will be heavy, especially with low wage increases and hence low tax revenues from newly employed youth. The level of employment will remain highly dependent on #1.
#3- The US government did not use the crisis as an opportunity to kick-start the development of the green economy, among other sectors, effectively missing any future chance of doing so if the economy recovers. This will keep the US behind the curve on the matter, reducing employment potential for #2, for decades to come, unless some president suddenly ends up with the 1930s or Cold War-era justification to massively invest in these sectors. Times have changed, so it is highly unlikely that a president would get such an opportunity again in the next ten years, at least. There will be no "Invasion of the Killer Oil Barrels" to warrant massive spending in a green economy, especially not with the way politics are played in the US.
#4- The banking system was not reformed, it will continue to act in a way that threatens lasting economic prosperity for short term gains, especially due to the likely backing of the government during downturns.
I could go on.
The idea that improvements in Europe's economic situation would significantly help the US recovery is a big joke, this is Obama trying to save his campaign. The US' problems are first and foremost related to its inaction internally, mainly lack of proper spending. If improvements materialized due to European economic growth, it would only benefit the US in a way that would not safeguard US economic growth and stability over a decent period of time.