During his meeting with international observation teams on Monday, Abbas, who wants to resume peace talks with Israel, expressed hopes that his offer would be acceptable to Tel Aviv.
The Palestine Liberation Organisation chairman won 62.3% of the vote in Sunday's election to succeed the late Yasir Arafat.
Abbas said: "We are ready to make peace based on justice and we hope that the response will be positive." But he acknowledged that there was a lot to do.
"The elections are only the beginning, not the end. We have a programme full of things to do in order to achieve peace," he said.
But on Tuesday, Israeli occupation troops assassinated a resistance activist from al-Aqsa Martyrs Brigades of the Fatah movement in the West Bank town of Tulkarim.
Aljazeera's correspondent said an Israeli special unit ambushed Sanad Ghannam in the northern suburb of Tulkarim, killing him on the spot. Two Palestinian passers-by received bullet wounds.
Sharon aides say he will seek a
meeting with Abbas soon
Israeli Prime Minister Ariel Sharon said on Monday peacemaking could start only with a crackdown on resistance fighters.
"The main focus at this stage should be Palestinian action on terror," he said.
"He [Abbas] will be tested by the way he battles terror and acts to dismantle its infrastructure."
Muhammad al-Hindy, a leader in the Islamic Jihad movement, told Aljazeera: "It is imperative for Fatah to pose as a resistance movement and to refrain from responding to the intense pressures that call for solving what Israel and the US describe as the question of terrorism."
Israeli officials have said Sharon, who accused Arafat of instigating violence and refused to deal with him for years, will seek a meeting with Abbas soon.
But Abbas aides said he wanted assurances that it would be more than just a photo opportunity.
"We are ready to
make peace based on justice and we hope that the response will
President, Palestinian Authority
On the other hand, US President George Bush, after years of shunning Arafat, wasted no time in contacting Abbas. In a 10-minute telephone call, Bush issued an open invitation to the new Palestinian leader to visit the White House.
The US president told Abbas he was "committed to helping him tackle key issues, like security, terrorism, economic growth and building democratic institutions", said White House spokesman Scott McClellan.
Abbas told Bush he was committed to addressing the Palestinian Authority's security and economic problems and looked forward to going to Washington at some point, McClellan said.
The US president wants both the Israelis and the Palestinians to implement a peace road map backed by the US, European Union, Russia and the UN.
Bush said earlier in the day: "It is essential that Israel keep a vision of two states living side by side in peace, and that as the Palestinians begin to develop the institutions of a state, that [Israelis] support the development of those institutions."
Abbas' victory margin was at the top end of expectations, but with the electoral commission declining to give a final turnout figure due to doubts over the number of eligible voters, the real strength of the mandate was difficult to gauge.
In a telephone call, George Bush
offered Abbas help on key issues
While the European Commission and other international bodies welcomed the outcome of the Palestinian elections, some Palestinians have been more cautious.
Groups, including the Islamist movement Hamas, boycotted the election and some fighters have defied Abbas' calls to end attacks on Israel.
Hamas cast doubt on the turnout size. Mushir al-Misry, a Hamas spokesman, said balloting time had been extended to tamper with elections since the voter turnout was very low.
"Abu Mazin has
explicitly stated that
he will never sign any agreement that does not incorporate freeing all Palestinian detainees and prisoners of war"
Islamic Jihad spokesman
But Islamic Jihad's al-Hindy told Aljazeera: "Election turnout was fully up to expectations. The intifada has helped Fatah to reclaim its strength and domination, particularly after the formation of al-Aqsa Brigades.
He said: "Around 60% of the voters took part in the elections and 65% of them voted for Abu Mazin. About 35% of the Palestinian public had endorsed him."
Al-Hindy said Islamic Jihad was awaiting clarification on Abu Mazin's domestic manifesto before committing their cooperation.
He said: "We will be waiting until Abu Mazin's programme is made clear to us. But based on our observations, Abu Mazin has explicitly stated that he will never sign any agreement that does not incorporate freeing all Palestinian detainees and prisoners of war."