US Secretary of State Hillary Clinton said she looked forward to working with him in "any new capacity".
Mrs Clinton said she and Mr Abbas had discussed his political future during her visit to the Middle East last week.
"He reiterated his personal commitment to do whatever he can to achieve a two-state solution to the Israeli-Palestinian conflict... I look forward to working with President Abbas in any new capacity in order to help achieve this goal," she said.
Speaking in Ramallah, Mr Abbas announced that he had no intention of standing in the presidential election scheduled for 24 January in the West Bank and Gaza.
Born in Safed in British Mandate Palestine (now northern Israel) in 1935; studied law in Egypt and gained doctorate in Moscow
A founder member, with Yasser Arafat, of Palestinian political faction Fatah
Held security role within the PLO in the early 1970s
Appointed head of the PLO's department for national and international relations in 1980
Widely regarded as an architect of the 1993 Oslo Peace Accords
In January 2005, elected president of the Palestinian Authority
"I have informed brothers in [the PLO and Fatah] that I do not wish to present my candidacy in the forthcoming presidential election," he said.
"This decision does not at all amount to bargaining or political manoeuvring... It is worth noting that I shall take other steps when the time comes."
The 74-year-old leader accused the US of backtracking on its Middle East policy and refusing to persuade Israel to freeze the construction of Jewish settlements in the occupied West Bank.
"We were optimistic when [US] President [Barack] Obama announced the need for a complete halt to settlements including natural growth," he said. "We were surprised by his [later] support for the Israeli position."
Nevertheless, Mr Abbas said Washington still had a pivotal role to play in eventually achieving peace in the region, which he was confident would happen.
"The two-state solution with Israel and Palestine living side-by-side in peace and security is still possible," he added.
The BBC's Middle East editor, Jeremy Bowen, says the decision is another serious blow to the US government's unravelling Middle East policy.
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