He had earliler ordered for elections to be held on January 24.
But the Central Election Commission announced last week it had advised Abbas to put off the election because Hamas, which rules the Gaza Strip where some 1.5 million Palestinians live, had warned it would not allow them to vote.
Hamas, which seized control of Gaza in 2007, disputes Abbas's legitimacy.
Abbas’s announcement reflected frustration with the stalled peace process and what the Palestinians see as the failure of the United States to put pressure on Israel to halt settlement activity on land occupied in the 1967 Middle East war.
The outgoing president insisted Israel must halt settlement activity before any resumption of talks.
"I said that the Israeli government does not want peace. The American government has not done enough for the sake of peace," he said.
"Now for a realistic reason, due to certain conditions - because of the rejection of Hamas and its threat to prevent (voting) by force, naturally they will be delayed, or the time of the elections will come later," Abbas said.
"It is better for us that Hamas accepts the holding of elections. But if that doesn't happen, then the Palestinian leadership must take measures."
Senior members of the Palestine Liberation Organisation (PLO) which Abbas heads, told Reuters this week they expected the body to effectively extend Abbas's term at a meeting in December.
The PLO and the Fatah faction which dominates it have both called on Abbas to stay on as leader.
Abbas called the January 24 vote after Hamas refused to sign an Egyptian proposal that would have scheduled elections for June.
Hamas says it has reservations about the Egyptian proposal, which aims to promote reconciliation between Fatah and Hamas.