NICOSIA, Cyprus — Europe’s surprising decision early Saturday to force bank depositors in Cyprus to share in the cost of the latest euro zone bailout set off increasing outrage and turmoil in Cyprus on Sunday and fueled fears that the trouble will spread to countries like Spain and Italy.

Facing eroding support, the new president, Nicos Anastasiades, asked Parliament to postpone until Monday an emergency vote on a measure to approve the bailout terms, amid doubt that it would pass. The euro fell sharply against major currencies ahead of the action, as investors around the world absorbed the implications of Europe’s move.

In an address to the nation, Mr. Anastasiades painted an apocalyptic picture of what would happen if Cyprus did not approve the strict terms: a “complete collapse of the banking sector”; major losses for depositors and businesses; and a possible exit of Cyprus from the euro zone, the 17 countries that use the euro as their currency.

He said he was working to persuade European Union leaders to modify their demands for a 6.75 percent tax on deposits of up to 100,000 euros, a move that would hit ordinary savers.