"The (poorer countries) are borrowing money on Northern Europe's credit card," he said during a panel discussion at the International Economic Forum of the Americas.
The system is not viable and continuing to bail out countries like Greece, Portugal and Spain won't work because the competitive position of these members will continue to erode relative to Germany, Greenspan said.
"The only solution to this is political consolidation of Europe," said Greenspan, who led the U.S. central bank from 1987 to 2006.
Greenspan said the "noble experience" of the Euro clearly demonstrates that culture really matters, pointing to the lower productivity in southern Europe compared with their northern neighbours.
He said these countries also have weaker savings rates, higher inflation and larger shadow economies that evade taxes.
Despite the additional costs of funding bailouts, Germany wants to preserve the euro because its low value ensures the country's exports are booming. But Greenspan isn't sure how long the central currency can last.
"Yes (the euro) has failed, the question is what happens now and what do we do about it," he said.
He said the European Central Bank can't continue to print money to bail out weak banks because the countries have very large deficits relative to the GDP.
The former central bank chief also expects the United States will face another debt ceiling crisis before a solution is found.
"There is a rapidly growing understanding in the United States political system that this is happening and it is a disaster for everybody, so I think we're going to get around that but not without some pushing and shoving and some angst."
Greenspan rejected calls by those who want even more stimulus to help the struggling U.S. economy.
"In my judgment, what would stimulate would be to do less and calm the system down and get the activism that is involved in creating uncertainty at a lower level," he said.
Around 100 people gathered outside the downtown hotel to protest spending cuts and fee hikes brought in by the Quebec government, including the tuition increases that have lead to a four-month long student strike.
The size of the protest equalled the number of police officers keeping a close eye on the demonstrators.
But that didn't prevent a festive atmosphere from taking over the crowd, which banged on pots and pans in the street below the hotel where Greenspan was speaking.
"Given that it is the economy that runs the world, we shouldn't be surprised when the people descend into the streets to say they don't agree with that way of doing things," said Denise Ouellette, a student taking part in the protest.
"Our elected officials aren't governing; they've sold out to finance."
Earlier, Canadian Trade Minister Ed Fast said he hopes to conclude the world's largest free-trade agreement with the European Union this year.
Agricultural policies involving Canada's dairy and poultry farmers are believed to be one of the key sticking points to reaching deals in Europe and the country's attempt to join the Trans-Pacific Partnership talks involving Asia-Pacific.
"In both cases we're having open and frank discussions about agriculture, about supply management, but we've also made some pretty clear commitments to our farmers and we're certainly not going to break those commitments," he said in an interview.
Canada has pushed hard for admission to the TPP since Prime Minister Stephen Harper indicated his formal interest in joining the club during a summit in Honolulu last fall.
However, the group of nine countries involved in the talks, including the U.S., Australia, New Zealand, Chile, Peru, Vietnam, Malaysia, Singapore and Brunei, have so far refused.
Fast said free and open trade, rather than protectionism, is the best way to secure global recovery and create jobs.
He singled out foreign countries such as Argentina, along with critics in Canada who have criticized efforts to expand Canada's free trade deals.
"We're disappointed with the steps (Argentina has) taken in trying to raise protectionist measures within their economy. I don't believe those measures are going to be sustainable in the long run."
Since 2006, Canada has concluded free trade agreements with nine countries — Colombia, Jordan, Panama, Peru, the European Free Trade Association (Iceland, Liechtenstein, Norway and Switzerland) and Honduras.
It is also deepening trade ties with the world's fastest growing markets in the world in such countries as Brazil, China and India.
— With files from Jonathan Montpetit