"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.
Mr 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.
Mr 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."
Hamas rejects recognition
Recognition of Israel and renunciation of violence have been among key conditions set by the international community 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 Mr Abbas, have agreed to set up a unity government, based on a national reconciliation document drawn up in June which implicitly recognises Israel.
However, Hamas leaders say the accord does not include recognition of Israel and Mr 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.
Mr Abbas expressed regret that the "roadmap" 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 Israel's construction of a security barrier in the West Bank, colonisation 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."
Mr 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."