“It’s high time for this settlement occupation to end now,” Mr Abbas told the 193-member UN General Assembly.
Israeli foreign minister Avigdor Lieberman blasted Mr Abbas following the speech, accusing him of “diplomatic terrorism”.
The world body overwhelmingly gave the Palestinians de facto statehood in 2012 by upgrading their UN status to “non-member state” from “entity”.
In his address, Mr Abbas also accused Israel of waging a “war of genocide” in Gaza and claimed Palestinians faced a future in a “most abhorrent form of apartheid” under Israeli rule.
“In the name of Palestine and its people, I affirm here today, we will not forget and we will not forgive, and we will not allow war criminals to escape punishment,” Mr Abbas said in his 30-minute address.
His comments were also immediately condemned as offensive by the United States.
“President Abbas’ speech today included offensive characterisations that were deeply disappointing and which we reject,” said state department spokeswoman Jen Psaki.
She said such provocative statements are “counterproductive and undermine efforts to create a positive atmosphere and restore trust between the parties”.
Mr Abbas said the Palestinians and the Arab Group had been working on a draft UN Security Council resolution “to push forward the efforts to achieve peace”.
Abu Rdainah, a spokesman for Mr Abbas, said Ramallah wanted a council resolution to call for an end to the occupation within “two to three years”.
Security Council diplomats said on Friday that Jordan, one of 15 members, had not yet circulated that draft resolution on behalf of the Palestinians and the Arab Group.
The council has already been negotiating for several weeks another Jordan-drafted resolution on an Egyptian-brokered ceasefire in late August that ended a 50-day Gaza war between Israel and the Palestinian militant group Hamas.
Britain, France and Germany have proposed including in that draft resolution the creation of an international ceasefire monitoring and verification mechanism.
Diplomats say such a mission could be established by the UN or European Union.