Abbas, who heads the Fatah movement, said he would remain engaged in the ongoing talks with the rival political faction Hamas in Egypt to achieve a viable Palestinian government.
"We have reiterated ... our determination to achieve the peace process according to the international legal road map, the two states solution and our full commitment to achieving this process," Abbas said.
"We will seek to hold parliamentary and presidential elections by January 24, 2010.
"The incoming Israeli government ... [must] respect the road map and two-states solution and should stop all settlement activity and re-open the border crossings [into Gaza]."
Shortly after the news conference an Israeli air strike left two Palestinians dead in Jabalya, Gaza.
The two men, who were members of Islamic Jihad, were travelling on a motorbike at the time of the strike. Several other people were wounded.
Marwan Bishara, Al Jazeera's senior political analyst, said: "We see an approach by the secretary of state that is completely noncommittal to anything specific [about the demolitions], which previous administrations saw as not only unhelpful, but also as illegal under the fourth Geneva convention.
"[Clinton] giving a short lecture on hope but really delivering nothing of substance to give hope, was a bit worrisome I'm sure for lots of her Palestinian listeners.
"The road map came out in 2002 under the Bush administration ... that time schedule is over four or five years ago, so the road map has already come to a dead end.
"It is peculiar to speak about it [the road map] as if it is something to refer to, rather than referring to international law ... or what [ to do] now after 15 years of peace processes that basically went nowhere ... not only do they have nothing new to say, but they are leaning on something which is passé."
Signs 'not positive'
Osama Hamdan, a senior Hamas official, said: "There is no change in the United States policy, the signs are not positive. Achieving any kind of peace process or at least a political process in the region [they] need to talk to the main players, mainly in the Palestinian situation.
"What she [Clinton] said is not important when you talk about the whole Palestinian issue. Everyone knows that the [Israeli] settlements are being enlarged, they are taking Palestinian land ... she criticised part of this, but she did not talk about achieving a peace agreement between the Palestinians and the Israelis. She did not talk about the new change in the Palestinian situation after the war in Gaza.
| Clinton spoke about hope in press conference following meeting with Palestinian students [EPA]
"They [the US] are not ready to deal with this reconciliation [between the Palestinian Authority and Hamas]. They are talking about a government without including Hamas."
The primary conditions of the road map for peace include ending Palestinian violence, an Israeli withdrawal from the occupied Palestinian territories and a freeze of Israeli settlement building in the West Bank.
Jerusalem's municipality issued orders on Tuesday to demolish 55 homes in Ras Khamees in Shu'fat Camp, in the centre of Palestinian East Jerusalem.
The municipality said that the homes were built without licences, whereas rights groups have said that the move is a new episode in Israel's campaign of collective eviction against Palestinians.
At the Egyptian resort of Sharm al-Sheikh on Monday, Clinton pledged the additional aid to help rebuild the Gaza Strip after Israel's 22-day offensive in the territory in December.
About 1,300 Palestinians and 13 Israelis died during the war, before unilateral ceasefires were declared by Israel and Palestinian factions in the Gaza Strip.
On Tuesday, Clinton urged Palestinians to "break the cycle of rejection and resistance" after meeting Israeli leaders in Jerusalem.
After her meetings in Jerusalem, Clinton emphasised the need for rocket fire from the Gaza Strip into Israel to end, and defended Israel's right to respond to such attacks.
The statement raises the possibility of disagreement with Benyamin Netanyahu, the Israeli prime-minister designate, who has said in the past that the Palestinians are not ready for statehood.
Netanyahu is currently attempting to assemble a ruling coalition, likely to be made up for hard-right and religious parties, after elections last month.
Though Obama has said that an Arab-Israeli peace deal will be a priority during his presidency, talks between Israeli officials and the Palestinians have stalled over violence, settlement-building and disputes over other core issues such as the future of Jerusalem and Palestinian refugees.
The PA suspended the negotiations after Israel launched its offensive on Gaza, with a stated aim of ending Palestinian rocket fire into the south of the country.