* U.S. debt ceiling doubts eroding confidence - EU's Barroso
* Sustainable global recovery depends on U.S. solution
* European Commission president urges rebuilding of EU-US trust
BRUSSELS, Oct 30 (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. But the temporary fix has only pushed the problem into early 2014, with no comprehensive solution in sight.
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.
"What is at stake is fundamental, not only for the American economy, but also for the credibility of the United States in the world," he told Reuters in an interview.
"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.
Former Treasury Secretary Timothy Geithner flew to Europe to meet EU finance ministers, and President Barack Obama sat down with a handful of leaders, including Barroso and German Chancellor Angela Merkel, on the sidelines of a G20 meeting to try to understand the depth of the region's challenges.
Two years on, the EU has managed to stabilise the situation, but only after five euro zone countries received rescue packages of one form or another, a 500-billion-euro rescue fund was established and much stricter fiscal rules were agreed among the 17 countries that share the single currency.
Now it is the United States, where elements within the Republican party are demanding that Congress not give in to demands of the White House and Treasury to raise the debt ceiling, that is causing consternation across Europe.
"One of problems that (our) crisis highlighted, and a lesson we should draw, is the issue of confidence. We have a serious problem of confidence," Barroso said.
"Investors globally have been risk averse and this is not good for a scenario where we want globally to restore sustainable growth.
"Everything that casts a shadow over the confidence of investors is certainly not good, neither for the country concerned, in this case the United States, nor for the global economy."
SPYING UNDERMINES TRUST
The uncertainty that exists between the world's two largest economies comes on top of the serious questions raised by the U.S. spying scandal, in which the National Security Agency is accused of eavesdropping on France, Spain, Italy and Germany, including Merkel's own personal mobile phone.
European leaders have expressed collective outrage at the U.S. activities, and in a phone call with Merkel last week, Obama assured her no such spying was going on any longer.
Germany and France are now in discussions with the United States over a new espionage relationship, including the possibility of a "no-spying" agreement. Obama is now considering a ban on any U.S. espionage against allies.
Barroso described the spying on Europe as a "very big surprise" and said concrete steps needed to be taken to rebuild trust with between the transatlantic partners.
"It's a matter of the utmost sensitivity. We believe it should be addressed in a way that gives reassurances that can build trust between the United States and Europe," he said.
"There is no more important relationship for Europe than the relationship with the United States, and I also believe there is no more important relationship for the United States than the one it has with Europe."
(Writing by Luke Baker)