"I would like to reaffirm that any future Palestinian government will commit to all the agreements that the Palestine Liberation Organization and the Palestinian Authority have committed to," he said in a speech to the assembly.
Abbas referred to letters exchanged by the late Palestinian and Israeli leaders, Yasser Arafat and Yitzhak Rabin, in 1993, which contained historic mutual recognition statements.
Abbas, who has been negotiating with the Hamas group over a national unity government, also said: "Any future government will commit to imposing security and order, to ending the phenomena of multiple militias, indiscipline, and chaos, and to the rule of law."
Recognition of Israel and renunciation of violence have been among key conditions set by the international community - led by the diplomatic Quartet on the Middle East - to assist any Hamas-led government in the Palestinian territories.
Hamas currently rejects both conditions.
Hamas, which won landmark elections in January, and Fatah, the party led by Abbas, have agreed to set up a unity government, based on a national reconciliation document drawn up in June that implicitly recognizes Israel.
Hamas leaders say, however, that the accord does not include recognition of Israel and Abbas froze negotiations with the rival group before leaving for the UN General Assembly where the Middle East peace process has come under a renewed spotlight.
In reaction to Abbas' announcement, an aide to Hamas Prime Minister Ismail Haniya said Friday that the next Palestinian government would not recognize Israel but was instead prepared to back a 10-year truce with the Jewish state.
Israel promptly rejected the truce suggestion and demanded that the Palestinian government recognize the Jewish state. "This does not interest us," government spokesman Avi Pazner said.
The UN Security Council held a special ministerial meeting Thursday to discuss the Israel-Palestinian conflict, at the request of the Arab League.
At the meeting UN Secretary-General Kofi Annan called for renewed efforts to bring together Israelis and Palestinians.
"The time has come to rebuild the shattered bridge to peace" between Israel and the Palestinians, Annan told the gathering. "Our continued failure to resolve this conflict calls into question the legitimacy and the effectiveness of this council itself."
There was no immediate US reaction to Abbas' speech, but after the Security Council meeting US Secretary of State Condoleezza Rice said: "We have welcomed his [Abbas'] efforts to bring an end to the crisis through the formation of a unity government.
"But a unity government would need to reflect the Quartet principles because you can't have peace unless you recognize the other partner for peace and renounce violence." The Quartet has insisted that any Palestinian government must recognize Israel, renounce violence, and stick to past accords made with Israel.
Rice also confirmed that she would visit the Middle East "fairly soon" as part of accelerated efforts to revive the stalled Israeli-Palestinian peace process.
In his speech, Abbas warned against leaving the Middle East conflict to fester.
"Without resolving the question of Palestine, and the continuation of the occupation of Palestinian and Arab lands since 1967, the elements of tension and conflagration will keep the conflict alive and will keep the door widen open to all forms of violence, terrorism, regional confrontations and global crises," he told the General Assembly.
Abbas expressed regret that the "road map" to peace approved by the Quartet - the United States, Russia, European Union, and United Nations - in 2003 had "reached a state of stagnation and even regression."
He said that Israel's construction of a security barrier in the West Bank, colonization and military raids help "despair and frustration thrive."
"Under these conditions, I can legitimately ask how the international community can expect extremism to retreat, or the waves of violence to ebb."
Abbas finished his speech by quoting a phrase uttered by Arafat when he spoke to the UN General Assembly in 1974, with a pistol holster around his waist and an olive branch in his hand: "Do not let the olive branch fall from my hand, do not let the olive branch fall from my hand."
© 2006 Agence France-Presse