The Dow Jones industrial average was down almost 200 points at its low point for the day before recovering most of its loss to finish down 76. It was the average's fifth straight decline.
European indexes closed near their lowest levels in months, and the euro neared a five-month low against the dollar.
Prices plummeted for commodities like oil and copper that depend on the health of the world economy. The turmoil in Europe added to concerns about slower economic growth in China and weaker job creation in the U.S.
Trading throughout the markets is growing more volatile as Europe's debt crisis "accelerates to a point where it's not really controllable with the sorts of Band-Aids they've used," said Daniel Alpert, managing partner at the investment bank Westwood Capital Partners LLC.
In Greece on Tuesday, the left-wing politician struggling to form a new government declared that the country was no longer bound by its promises cut spending sharply in exchange for international bailout loans.
The politician, Alexis Tsipras, also demanded a moratorium on repaying the part of Greece's debt that is "onerous." The main stock index in Greece closed down 3.6 percent after a 7 percent decline the day before.
Greek voters on Sunday rejected parties that had imposed the deep spending cuts demanded by Greece's bailout lenders. Cuts to pensions and social programs are deepening Greece's crushing recession.
After a calm finish Monday, benchmark indexes in Germany and France plunged to near their lowest levels this year. Italy's was near its lowest since last November. The main stock index in Britain hit its lowest point this year.
Central banks have injected billions into Europe's financial system, providing temporary support for stock and commodity prices, Alpert said. "If that liquidity is supposed to prime the pump, and the pump doesn't take over, then you've got a problem," he said.
In the U.S., traders dumped risky assets and commodities, partly because of concern that a punishing recession in Europe would hurt demand for U.S. exports. The price of oil continued its week-long slide. Copper and silver each lost more than 2 percent.
Gold fell to a four-month low, dipping below $1,600 per ounce for the first time since early January. Gold often serves as a safe, stable investment to hold in turbulent times. But in periods of rapid selling, investors sometimes sell gold as a ready source of cash.
The stronger dollar contributed to the fall in commodity prices. Commodities are priced in dollars, so a stronger dollar makes them appear more expensive to traders who use other currencies.
Money flowed into safe investments such as U.S. Treasurys, pushing the yield on the 10-year Treasury note down to 1.85 percent from 1.88 percent late Monday.
A flurry of late-day buying helped the indexes recover from their earlier lows. The Dow closed down 76.44 points, or 0.6 percent, at 12,932.09. The Standard & Poor's 500 index fell 5.86, or 0.4 percent, to 1,363.72. The Nasdaq composite index fell 11.49, or 0.4 percent, to 2,946.27.
Markets have been buffeted for three years by shifting perceptions about the gravity of the European debt crisis. At times, many feared a messy string of defaults would set off a global credit crunch.
To soothe bond investors and prevent nations' borrowing costs from rising, leaders of indebted nations agreed to spending cuts that hurt workers' standards of living. They slashed pensions and government jobs, raised retirement ages and eliminated social programs.
Opponents of strict austerity say Europe will be unable to emerge from its recession unless governments spend more to boost demand in the economy. French voters on Sunday elected a president who has spoken out against austerity and promised to cut France's debt load more slowly.
Uncertainty about Europe's path forward is injecting volatility into markets that were placid earlier this year.
The markets had a calm first quarter after international lenders and leaders of Germany, Europe's economic engine, helped push through austerity programs in several indebted nations.
Governments in Italy and Greece fell, replaced by technocrats whom international leaders trusted to navigate the crisis. To shore up the region's shaky banks, the European Central Bank injected billions of euros into the financial system.
As fears about Europe and the U.S. economy reemerged in recent weeks, traders have returned to frenzied buying and selling that recalls mid-2011's record-breaking volatility.
Some corporate news that moved stocks:
— Watchmaker Fossil plunged 37.6 percent after saying weak sales in Europe caused its first-quarter revenue to fall far short of expectations. The company also lowered its 2012 earnings forecast.
— Casino operator Wynn Resorts reported a disappointing drop in first-quarter earnings, sending its stock down 4.8 percent.