Abbas made the demands in a speech to the new Hamas-dominated parliament, after legislators were sworn in by collective oath. Hamas controls 74 of 132 seats, but Abbas retained considerable power as Palestinian Authority president, controlling foreign affairs, security and peace negotiations.
Hamas legislators quickly rejected Abbas' demands. Negotiations with Israel are "not on our agenda," said Mushir al-Masri, a leading Hamas legislator.
After the session, both Abbas and Ismail Haniyeh, Hamas' choice for prime minister, said they would try to resolve their deep differences through dialogue.
Haniyeh said he hoped a compromise could be reached, despite conflicting views. "We will deal with this difference in the political position through dialogue and understanding, to preserve the national unity of the Palestinian people and promote the higher interests of our people," Haniyeh said.
Hamas is pledged to Israel's destruction, opposes peace negotiations and has said it is not moderating its ways, despite international pressure and threats of tough Israeli sanctions, such as a blockade of the Gaza Strip.
Abbas has not yet decided whether he'd fire a Hamas prime minister who rejected his policies, Abbas aides have said. After leaving parliament Saturday, Abbas told reporters: "Why assume that there will be crisis? Let us resort to dialogue. Everything comes through dialogue."
Israeli government officials declined comment Saturday.
Israel considered Saturday's swearing-in ceremony as a turning point, after which it will drastically scale back ties with the Palestinian Authority. On Sunday, Israel's Cabinet is to vote on sanctions against the Palestinians, including sealing Gaza, cutting off Gaza from the West Bank and keeping out thousands of Palestinian workers.
Israel is threatening to withhold 50 million dollars in monthly taxes it collects from the Palestinians, reports CBS News Correspondent David Hawkins, and the U.S. says it will cut off all foreign aid.
Saturday's parliament session was held simultaneously in the West Bank city of Ramallah and in Gaza City, because of Israeli travel bans. Legislators were hooked up by video conference.
In Gaza, some 2,000 diplomats and other VIPs gathered in a government complex to view the parliament session, along with about 100 women from the Hamas Women's Union, their faces covered by veils. In Ramallah, Hamas lawmakers entered the main hall in Abbas' headquarters in a group, one carrying a picture of jailed lawmaker Hassan Yousef. The group's female lawmakers covered their heads in traditional Muslim fashion.
Abbas said in his opening speech that a new reality has been created by the Hamas victory in last month's parliament election and the defeat of his Fatah Party.
"Therefore, it (Hamas) will be asked to form the new government," Abbas said. "On my part, you will find all the cooperation and encouragement you need, because our national interest is our first and final goal, and is above any individual faction."
Abbas asked Hamas to put together the government as quickly as possible, and to name its candidate for prime minister. Hamas said its choice is the group's pragmatic Gaza leader, Haniyeh, who in the past served as a liaison with the Palestinian Authority. However, no formal announcement has been made.
Parliament elected a new speaker, Hamas legislator Abdel Aziz Duaik. The outgoing speaker, Rauhi Fattouh of Fatah, handed over the gavel, to cheers and applause from Hamas legislators.
In his maiden speech, the West Bank geography professor said Hamas will try to fulfill its "rightful duty to resist occupation." He also said the new parliament would review "all decisions and decrees" issued during the transitional period, an apparent reference to the last session of the old parliament in which Abbas was given additional powers.
Israel, the United States and the European Union, which provide funding for a majority of the Palestinian Authority's budget, have threatened to sever financial ties with the government if Hamas does not renounce violence and recognize Israel. Hamas is branded a terrorist group by the United States and Europe.
Israel was also considering a basket of sanctions against the Palestinian Authority, and its Cabinet will vote Sunday on measures meant to put a stranglehold on the Hamas parliament. The moves would devastate the already frail Gaza economy, though Israel would continue to allow humanitarian shipments to cross into the coastal area.