By Luke Baker and Stephen Adler BRUSSELS (Reuters) - Persistent doubts about the ability of the United States to resolve its debt problems are putting U.S. credibility in the world at stake, European Commission President Jose Manuel Barroso has warned. The White House won a brief respite this month when the Republicans backed down after a 16-day government shutdown. Barroso, who heads the European Union executive and has been at the frontline of efforts to resolve Europe's debt crisis over the past three years, said the uncertainty was making investors risk averse, with potentially damaging economic consequences. "I hope that American democracy will work and will deliver what I think is critically important, not only for America but also for the world, because of the size and the impact of the American economy in the world." At the peak of Europe's debt crisis, when it looked like Greece could be forced out of the single currency zone and global markets were on edge, U.S. officials repeatedly sought assurances from the EU that leaders had the situation in hand.