Mr Abbas made the comment during an interview with Britain's ITV network, in which he urged the international community to give Hamas -- classified as a terrorist group by Israel, the United States and the European Union -- a grace period.
"I think they are now responsible, and in order to assume responsibilities, their policies have to be compatible with international polices," he told interviewer Jonathan Dimbleby.
When asked if he would step aside if peacemaking remains elusive, Mr Abbas said: "We could reach a point where I cannot perform my duty. Then, I will not continue sitting in this place, against and in spite of my convictions. If I can do something I will continue, otherwise I won't."
Mr Abbas was elected president of the Palestinian Authority in January 2005 for a four-year term.
A power struggle between Mr Abbas and Hamas emerged when the radical Islamic party won control of the Palestinian parliament from the long-ruling Fatah Party last month.
Mr Abbas said despite its parliamentary majority, Hamas must accept his policies and abide by existing Palestinian Authority (PA) commitments recognising Israel and foreswearing the use of violence.
Hamas denies recognition
Adding to the tensions, Hamas' prime minister-designate, Ismail Haniyeh, has denied that he had suggested the radical group might one day recognise Israel.
Clarifying his comments to the Washington Post newspaper, Mr Hamas said that there was only a possibility of achieving a long-term truce.
Mr Haniyeh, a 43-year-old viewed by many Palestinians as a pragmatist, told reporters in Gaza that he "did not tackle the issue of recognising (Israel) in my interview with the Washington Post".
Reiterating a long-standing position by Hamas, Mr Haniyeh said the group would never recognise Israel but could agree to a long-term truce if Israel withdrew from lands captured in the 1967 war, freed prisoners and allowed the return of refugees.
The Washington Post newspaper on its website quoted Mr Haniyeh as saying in an interview: "If Israel declares that it will give the Palestinian people a state and give them back all their rights, then we are ready to recognise them."
Hamas, which seeks to replace Israel with an Islamic state, has carried out nearly 60 suicide bombings since a Palestinian uprising began in 2000.
But it has largely abided by a ceasefire forged a year ago.
Abbas irrelevant
Meanwhile Israel says it regards Palestinian President Mahmoud Abbas "irrelevant", calling Hamas the real governing power and urging the international community not to weaken its pressure on the militant group to seek peace.
The description of the moderate leader, who won public backing by a US envoy over the weekend, came on the eve of a European Union meeting on preventing the Palestinian Authority's financial collapse and possibly channelling aid to Mr Abbas.
Israel has stopped the monthly transfer of up to A$74.57 million in tax payments to the Palestinians.
Foreign donors, including the EU and the US, have threatened to cut off funds to the Authority once a Hamas-led government is sworn in.
Israeli Foreign Minister Tzipi Livni said she feared Mr Abbas, who seeks a negotiated peace with the Jewish state, could serve as a "fig leaf" for an administration dominated by Hamas.
"Our job, among other things, will be to ensure the international community does not ... embrace Abu Mazen or some moderate statements by Hamas," she said on Channel One television, using Mr Abbas's popular name.
"The ball is in the Palestinian court and it is up to the future Hamas government to do something with it, and in this context, Abu Mazen is irrelevant," Ms Livni said.
Chief Palestinian negotiator Saeb Erekat called Ms Livni's statement "absolutely unacceptable" and said it reflects Israel's "non-partner policy" that has led to "military escalation".