"We are entrusted and you are entrusted to end apartheid in Palestine," Abbas told the UN General Assembly in a nearly 45-minute address.
"Can the world accept an apartheid regime in the 21st century?" he asked.
Taking the podium a day after Israeli Prime Minister Benjamin Netanyahu, Abbas slammed Israel over the construction of new settlements "everywhere," saying they were putting the two-state solution in jeopardy.
"There is no place left for the state of Palestine and this is not acceptable," he said.
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The resolution passed after the United States under the previous administration of Barack Obama declined to use its veto and instead abstained.
The Palestinian leader vowed to push for full recognition of Palestinian statehood at the United Nations, a move that would require approval from the Security Council where the United States, Israel's key ally, holds veto power.
Abbas spoke at the assembly after meeting US President Donald Trump who said he was "working very hard with everybody involved toward peace" but offered little detail.