Abbas suggested the Palestinians can only hope to make progress on the U.S.-backed "road map" peace plan if he is given the full power to implement it.
However, reports CBS News Correspondent Robert Berger, Abbas stopped short of calling for a no-confidence vote that could topple his unpopular government.
Eighteen of the 83 legislators signed a petition calling for such a vote, and the parliament speaker, Ahmed Qureia, was to consider the request. Qureia had said Wednesday he opposed holding a vote, arguing that parliament should not be dragged into the struggle between Abbas and Arafat.
Outside, demonstrators accused Abbas of being an agent of the CIA.
The U.S.-backed Abbas, who has minimal support among Palestinians, could be toppled in a confidence vote, dealing a heavy blow to efforts to end three years of violence and move toward Palestinian statehood.
Earlier Thursday, Palestinian gunmen killed an Israeli in a shooting ambush near the West Bank town of Jenin. The Al Aqsa Martyrs' Brigade, an armed group linked Fatah movement claimed responsibility.
In his speech to parliament, Abbas only hinted at his conflict with Arafat, saying there were "problems" between his government and the Palestinian leadership. Backed by the United States, Abbas demands that Arafat relinquish control over security forces. For now, each leader commands four of the armed services, and Arafat has balked at giving up one of his last vestiges of power.
Abbas told parliament that "without a legitimate force in the hands of one authority ... we will not advance one step on the political track," a reference to the road map. The United States and Israel demand that the Palestinians begin dismantling armed groups, as required by the road map.
However, Abbas also told parliament Thursday that he had no intention of ordering a clampdown. "This government does not deal with the opposition groups with a policing mentality, but with a mentality of dialogue," he said.
In his 18-page speech, Abbas portrayed a unilateral cease-fire, declared by the armed groups June 29, as the main achievement of his first 100 days in office. He accused Israel of having sabotaged the truce with deadly arrest raids to try to evade its obligations under the peace plan.
Militants carried out reprisal bombings for those arrest raids, including one Aug. 19 in which 21 people were killed on a Jerusalem bus. Israel, in response, killed a senior Hamas leader in a targeted missile strike, prompting Hamas and Islamic Jihad to formally call off the truce.
"I don't think that anyone in the world would disagree with me when I say that Israel is responsible for the situation," Abbas said.
He said the United States did not do enough to stop what he referred to as "Israeli provocations" during a period of relative calm.
Abbas said he is leaving his political future in the hands of parliament.
"I am not attached to this post and I am not and will not make any effort to keep this post. It is a difficult mission that many describe as impossible," he said.
He ended his speech with an appeal for support, saying parliament must either back him or send him home. "You either provide the resources of power and support those things ... or you take it back," he said.
Winning parliament's support would help Abbas in his confrontation with Arafat, who is accused by Israel of fomenting terrorism. Defeat would allow him to step down without being blamed for the consequences, such as the possible collapse of the road map.
At the start of the session, about 200 activists in Arafat's Fatah movement demonstrated in support of the veteran Palestinian leader, and seven masked men from the crowd broke down a door to the building and smashed windows. Unarmed guards eventually forced the men out.
In Washington, meanwhile, Secretary of State Colin Powell reacted harshly to a comment attributed to Arafat that the road map is dead because of what the Palestinian leader described as Israeli aggression. Powell said Arafat "has not been playing a helpful role."
"If he wanted to play a helpful role he would be supporting Prime Minister Abbas, not frustrating his efforts," Powell said.
Israel has warned of dire consequences should Abbas be ousted, saying it will not do business with a government hand-picked by Arafat. Several Palestinian legislators said they were told by local U.S. diplomats that if Abbas is toppled, Washington might lower its profile as Mideast mediator.