ATHENS—Greece shook global markets, intensifying fears of a default, as tens of thousands of demonstrators protested a new round of budget-cutting plans and its prime minister offered to step down to try to preserve them.

Protests across the capital sometimes turned violent as Prime Minister George Papandreou sought an agreement with opposition parties on austerity measures demanded as the price of a new bailout by euro-zone nations and the International Monetary Fund.

When...