The World Bank's 24-member board will ultimately decide what action, if any, to take.
"If he won't jump himself, he must be pushed"
Member of European parliament
The controversy has prompted calls for Wolfowitz, a former member of the Bush administration, to step down from many of the bank's staff, former World Bank executives, aid groups and some Democrat politicians.
They fear the matter has tarnished the reputation of the institution, which is responsible for fighting global poverty, and could hobble efforts to raise billions of dollars to bankroll a World Bank programme to help poor countries.
The EU assembly said it wanted to "signal to ... Wolfowitz that his withdrawal would be a welcome step toward preventing the bank's anticorruption policy from being undermined."
They voted 333-251 with 31 abstentions to include a paragraph in a resolution on transatlantic relations calling on Germany, holder of the 27-nation bloc's rotating presidency, and the United States to ask Wolfowitz to stand down.
"By digging in his heels and refusing to resign as president of the World Bank, Wolfowitz is dragging the whole organisation into disrepute and further undermining the credibility of its anti-corruption policy," said Caroline Lucas, a British member of the Greens.
"If he won't jump himself, he must be pushed."
Wolfowitz has acknowledged making a mistake and has apologised but has said he will not resign.
At the White House on Wednesday George Bush, the US president, took care to offer a positive mention of Wolfowitz at an event to mark Malaria Awareness Day.
"I appreciate very much the fact that the World Bank is taking the lead in eradicating poverty in places like Africa, and Paul Wolfowitz, thank you for your leadership of the World Bank,'' Bush said.
Wolfowitz was deputy defence secretary in the first Bush administration and helped plan the 2003 US invasion of Iraq.
The United States - the bank's largest shareholder - has expressed confidence in Wolfowitz, but representatives of European governments want to see him go.
Hoping to repair relations with staff and boost morale, Wolfowitz said on Tuesday he would make "major changes" in the way his office and the senior management team work in order to deal with the concerns.