A senior Hamas leader has rejected demands that the Islamic militant group must renounce violence to prevent aid cuts for the Palestinian Authority.
Fatah supporters are unhappy with their leadership
Ismail Haniya, who headed Hamas' election list, said they would not give in to "blackmail" by foreign donors.
President George W Bush warned US aid, worth $400m (£225m), could be cut following Hamas' surprise poll win.
In fresh unrest, gunmen from the former ruling party Fatah climbed on to the Palestinian parliament and fired shots.
Hamas won 76 of the 132 seats in the Palestinian assembly and has the backing of a further four independent MPs.
Following the victory, the main donors to the Palestinian Authority, which has always been heavily reliant on international cash, said they were reviewing their funding position.
Mr Bush said the US would freeze aid unless Hamas renounced violence and stopped calling for Israel's destruction.
"If they don't, we won't deal with them," he told US TV channel CBS News. "Aid packages won't go forward."
Washington is reviewing all money given to Palestinians through the UN or non-governmental groups.
But Mr Haniya told the Reuters news agency: "This aid cannot be a sword over the heads of the Palestinian people and will not be material to blackmail our people, to blackmail Hamas and the resistance. It is rejected."
Hamas is classified by the US and the EU as a terrorist group.
The EU, the Palestinian Authority's largest donor with $606m (£341m) a year, has so far not threatened direct aid cuts.
But a French diplomat quoted by Reuters said a refusal by Hamas to give up violence would have consequences.
Israel has cast doubt on its willingness to continue to transfer customs and tax revenue, worth $50m, collected on behalf of the Palestinian Authority.
A senior Israeli defence ministry official also said on Saturday it would not grant Hamas legislators free access between Gaza and the legislature in Ramallah the West Bank, making the governing of the divided territories more difficult.
The supreme leader of Hamas, Khaled Meshaal, who lives in exile in Syria, said on Saturday he had been in contact with Mr Abbas and wanted "a partnership formula" for the new administration.
He said Hamas had succeeded in its resistance and would now succeed in "politics, reforms and change".
Hamas leader Khaled Meshaal says it will continue its "resistance"
However, unrest continues between Hamas supporters and members of Fatah and the security forces.
The worst of the violence was in Khan Younis in Gaza, where several people were wounded when rivals exchanged fire.
Also in Gaza, police loyal to Fatah briefly occupied a parliament building to protest against any transfer of security responsibility to Hamas.
Fatah members also turned on their own leadership, with demonstrators storming the legislative building in Ramallah to demand the resignation of Fatah heads. Parliament was not in session at the time.
Talks are expected soon between Hamas leaders and Mr Abbas on the new administration.
The BBC's Jim Muir in Gaza says Hamas is being coy about whether it intends to form a new government.
One idea being floated, he says, is for Hamas to support a neutral government of technocrats and to focus on social and economic issues.
The US, UN, EU and Russia 'quartet' working for peace in the Middle East will hold talks in London on Monday to discuss the Hamas victory.