Latin American stocks, meanwhile, rebounded strongly in tandem with a recovery on Wall Street.
The Bush administration says the money would be used to buy up soured loans and free banks to resume lending. That is seen as a key to economic growth in the United States, a major trading partner around the world.
In New York, the Dow Jones industrial average finished up 485.21 points, or 4.68 percent, to 10,850.66, gaining back some of its 7 percent loss on Monday, when it plunged 778 points in its biggest single-day point drop.
Britain's benchmark stock index, the FTSE 100, closed up 1.7 percent after falling by as much as 3 percent earlier in the day. Germany's benchmark DAX index rose by 0.41 percent, while the Paris CAC-40 ended up 2 percent.
Russia's regulator, meanwhile, was forced to halt regular trading for two hours in its two major markets on Tuesday morning after stocks plunged in the opening minute of trading. But shares recovered and closed up for the day.
In Ireland, the volatility was massively upward, as the government guaranteed all the deposits and borrowings - worth around 500 billion euros ($717 billion) - of six of the country's major lenders. Ireland's ISEF Index of financial shares surged by as much as 25 percent on the back of the guarantee, before settling to a rise of 7.9 percent.
Some analysts were crediting Ireland's unprecedented move with helping to keep European stocks overall from falling nearly as much as stocks in the U.S. and Asia had.
"The Irish government's blanket insurance could form a template for a European approach to this crisis," said Rob Carnell, London-based chief international economist at ING Financial Markets.
The high volatility was bad for credit markets, which became even more paralyzed by some measures. The rate banks charge each other for overnight dollar loans, the London interbank offered rate, or LIBOR, soared to an all-time high of 6.875 percent on Tuesday, indicating that banks are unwilling to lend to one another.
"The credit markets really need a U.S. bailout deal to go through," said Richard Hunter, head of British equities at Hargreaves Lansdown Stockbrokers. "The alternative is unacceptable."
In Asia, most major stock markets fell Tuesday in stunned dismay over U.S. lawmakers' rejection of the bailout plan.
Markets across Asia tumbled sharply as they opened amid fears that the setback could lead to a broader global financial crisis. But as trading progressed, many markets recovered somewhat and Hong Kong's market managed to close slightly higher as investors scooped up beaten-down shares.
Japan's benchmark Nikkei stock 225 index slumped 4.12 percent to close at 11,259.86 - the lowest level since June 9, 2005. In Australia, the S&P/ASX-200 index fell 4.3 percent after falling as much as 5.3 percent.
The bailout bill's failure dealt a "severe blow to Asia markets right after the Lehman shock," said Mitsushige Akino, fund manager at Ichiyoshi Investment Management in Tokyo, referring to the collapse earlier this month of the U.S. investment bank.
"Many investors grew even more cautious because of the latest development over the bill, and they only see passage of the bill as a minor improvement to the crisis," he said.
Some markets bounced back. Hong Kong's Hang Seng index gained 0.76 percent to close at 18,016.21 after earlier plunging more than 5 percent. India's Sensex was up 2.4 percent in afternoon trading.
Investors were stunned by the U.S. House of Representatives' rejection Monday of a $700 billion emergency bailout package that would have allowed the government to buy bad mortgages and other sour assets held by troubled banks and other financial institutions.
With elections in November, many lawmakers were unwilling to take the political risk of supporting a measure that many American voters see as an undeserved bailout for rich, reckless investment bankers.
Japan's banks have relatively little exposure to the bad mortgages at the core of the global credit crisis, but investors are worried that a slowdown in the U.S. and global economy will hurt demand for exports.
The Bank of Japan on Tuesday morning pumped another 3 trillion yen ($28.7 billion) into money markets, as part of efforts by central banks worldwide to boost liquidity and bolster interbank lending. That brings the BOJ's total injection to 21 trillion yen ($200.6 billion) since the collapse of Lehman Brothers Holdings Inc. earlier this month.
Markets in mainland China are closed this week for National Day celebrations, and Hong Kong will be closed Wednesday.
Latin American stocks, however, regained ground Tuesday after steep losses triggered by fears that without a U.S. financial bailout there could be a global recession that would hurt their commodity-based economies.
Gains were greater than in any other region of the world, but didn't erase the beating that markets took Monday.
Sao Paulo's Ibovespa index rose 7.6 percent to close at 49,541 after falling 9.4 percent a day earlier, its steepest drop since 1999. Brazil's currency, the real, regained some ground against the U.S. dollar after hitting a one-year low on Monday.
Mexico's main IPC index closed 3.9 percent higher at 24,889, while Chile's Ipsa index rose 4.6 percent to 2,753 and Buenos Aires's Merval index climbed 3.4 percent to 1,598.
