He has also appealed to Israel and the international community not to push Hamas "into a corner" but give it time to moderate its long-held stance of refusing to recognise Israel and renounce violence before mounting any boycott.
Accepting that Hamas's position in power has to be "compatible with international polices", Mr Abbas praised the faction's "wise and rational" Prime Minister-designate, Ishmael Haniya, as "flexible and diplomatic."
Mr Abbas said, in an interview for ITV1's Jonathan Dimbleby programme, that Hamas must go "very clearly" in the direction of renouncing violence and recognising Israel. He also suggested that the faction's tour of Arab states and Russia was likely to be influential. He added: "They will listen to many things that will make them think about their political position. I think they are now responsible and, in order to assume responsibilities, their policies have to be compatible with international policies."
But asked if he would resign if he cannot deliver what he wants in terms of the peace process, Mr Abbas said: "We could reach a point where I cannot perform my duty - then I will not continue sitting in this place. If I can do something I will continue, otherwise I won't."
Mr Abbas underpinned his threat by pointing out that, in 2003 he resigned as Prime Minister after making a similar threat when he failed to persuade the then president, Yasser Arafat, to reorganise control of the security services.
Mr Abbas implied, in his first international interview since the Palestinian elections, that the immediate priority was to secure agreement from Hamas to abide by all previous agreements - including the Oslo accords, which recognise Israel - and the "road map" for peace. Israel and the international community have declared this condition is additional to that of explicit recognition and the renunciation of violence.
The prospect that he might go if he fails to achieve progress - with both Hamas and the United States having urged him to stay in the post after the faction's election victory last month - probably remains his strongest card. At present, Israel has frozen its monthly $50m (£28.7m) remittance to the PA of the duties it collects on its behalf in response to the election victory. The US is also carrying out an urgent review of its funding of the PA.
Mr Abbas also indicated a possible compromise on a key dispute with Hamas over who will control the security services when Hamas assumes power. Pointing out that the police and preventive security forces fall under the Ministry of the Interior while the national security forces and presidential guard are run by the President, Mr Abbas said: "All these security forces have to work in harmony and in coordination."
Mr Abbas said that Hamas hadindicated that it will offer a long-term truce in return for an Israeli withdrawal to 1967 borders. He compared the current negotiations with the diplomatic dance before the signing of the Oslo accord when the Palestinian Liberation Organisation (PLO), he said, went within less than a fortnight from being seen as a "terrorist organisation" to signing the Oslo accords in the White House.
Mr Abbas appeared to reinforce Israeli scepticism that Iran can either afford to fund the PA or to channel cash through the Israeli-monitored banking system. He said "two questions" were: "Can Iran respond to all the demands of the Palestinian people? And ... how will the money be channelled to the PA?"
The Jonathan Dimbleby programme will be broadcast tomorrow on ITV1 at 10.25am.Reuse content