The decision reflected lawmakers' reluctance to get dragged into the dispute between Abbas and Arafat over control of security forces and over the course of peace negotiations. Abbas is to address parliament Thursday to sum up his first 100 days in office, and he told a senior Palestinian mediator that he wants the debate to be followed by a confidence vote.
A "yes" vote would strengthen Abbas' position; a "no" vote would save him the embarrassment of resignation by forcing him out.
If Abbas forces on a vote, many legislators might not support him.
"Arafat has become a symbol," parliament member Ziad abu Ziad told the Israeli newspaper Haaretz. "Everyone will support Arafat, will raise him up like their elderly father whom they won't give up."
Abbas' ouster, in turn, could throw an already troubled U.S.-backed peace plan into greater turmoil, since Israel won't deal with Arafat, reports CBS News Correspondent Robert Berger.
Abbas has failed in his two key objectives — persuading militants to halt attacks on Israelis and improving the daily lives of Palestinians through progress on the "road map" peace plan.
Palestinian Parliament Speaker Ahmed Qureia said Wednesday that Abbas already won parliament's confidence in April when he was sworn in, and that there was no need for another vote. Qureia said parliament should not be dragged into the Arafat-Abbas struggle.
"Parliament will not accept being turned into a place of conflict or to be part of the current crisis," Qureia said.
A vote could still be arranged if more than half of the 83 legislators demand it, but this seems unlikely. Many lawmakers, while critical of Abbas, say they don't want to push him out and prefer that he work out his differences with Arafat.
The power struggle between Abbas and Arafat intensified in recent weeks. Abbas, with the backing of the United States and Israel, demands that Arafat relinquish control of four security branches; Abbas commands the other four security services. Arafat has balked, fearing he would lose his main source of power.
Israel and the United States want Abbas to crack down on Hamas and other militant groups, as required by the road map. Israel's Cabinet decided earlier this week to freeze implementation of the road map until Abbas orders a clampdown.
Abbas and his security chief, Mohammed Dahlan, have taken some measures, including freezing the bank accounts of Islamic charities with alleged links to Hamas, but it remains unclear whether they would order a full crackdown even if given control of all security services.
Ahead of the parliament session, Israel sent strong warnings to the Palestinians.
Israeli Foreign Minister Silvan Shalom said Israel will not negotiate with a new Palestinian government handpicked by Arafat, and Defense Minister Shaul Mofaz warned Tuesday that Israel may have to expel Arafat soon if he keeps getting in the way of the road map.
Mofaz spoke several days after Israeli security officials again reviewed a possible expulsion, and came to the conclusion that it would cause more damage than good. Over the weekend, Israel's vice premier, Ehud Olmert, said another option is to isolate Arafat completely at his headquarters in the West Bank town of Ramallah, as Israel has done in the past.
Zeev Schiff, a veteran Israeli military commentator, wrote Wednesday in the Haaretz daily that Israel might expel Arafat under three scenarios: Arafat refuses to relinquish control over the security services, Abbas is forced to resign, or militants carry out a major terror attack.
Qureia, the Palestinian parliament speaker, said the Israeli threats should not be taken lightly. "That's why we are saying that the crisis should reach an end," Qureia said.
Israel Army Radio reported that the United States has asked Israel for a clarification following Mofaz's expulsion comments.
An official at the U.S. Embassy in Tel Aviv, speaking on condition of anonymity, said Arafat is considered "part of the problem, not the solution," but that Washington does not want him expelled.
Several Palestinian legislators said privately that U.S. diplomats have warned them that the United States might withdraw as a mediator if Abbas is brought down.
Chuck Hunter, a spokesman for the U.S. consulate in Jerusalem, declined comment on whether phone calls have been placed to Palestinian legislators. "Having said that, Abu Mazen (Abbas) is someone the (U.S.) president can work with and looks forward to continuing to work with," Hunter said.
Arafat, meanwhile, was quoted as saying that the road map had no future. The plan, launched June 4 by President Bush, calls on Israel and the Palestinian Authority to end nearly three years of violence and envisions the establishment of a Palestinian state by 2005.
"The road map is dead, but only because of Israeli military aggression in recent weeks," the CNN Web site quoted Arafat as saying.
Palestinian legislator Saeb Erekat later clarified that the Palestinian Authority stands by the road map. "We want the road map to stay on the table, and we want the implementation of the road map," he said.