In Hebron, al-Aqsa Martyrs' Brigades fighters, an offshoot of Fatah, stormed government offices and set up checkpoints to search for Hamas members.
But in Gaza, Ahmed Bahar, the deputy parliament speaker and a Hamas member, called Abbas's attempt to form an emergency government illegal.
Fatah remains in control of the West Bank and Abbas is based in Ramallah, the Palestinians' administrative centre.
On Saturday, Fatah supporters in the West Bank attacked Hamas politicians and establishments to avenge the killings of Fatah members in Gaza.
Hamas later threatened to retaliate in the West Bank against Fatah - which is headed by Abbas - if it did not stop "terrorism" there.
Early on Sunday, the Israeli prime minister said the formation of the Palestinian government has created a "new opportunity" for peace moves.
Israel will recognise a Palestinian government that does not contain ministers from Hamas, Ehud Olmert said before departing for a visit to the US.
"A Palestinian government which is not a Hamas government is a partner and we will co-operate with it."
|Fayyad is an independent parliamentarian|
who has worked in the World Bank [AP]
In another boost for Abbas, a US envoy told Abbas during a meeting in the West Bank on Saturday that the US would lift a direct ban on aid to the government once the new administration was announced.
Jacob Walles, the US consul-general in Jerusalem, said: "I expect that we are going to be engaged with this government.
"I expect that early next week. There will be some announcements in Washington, specifically about our assistance and about the financial regulations."
The emergency government will have between 10 and 12 members and consist primarily of independent technocrats.
"Most crucial will be the position of the minister of interior who will be in charge of security in the West Bank, and the Gaza Strip if possible," Walid Batrawi, Al Jazeera's correspondent in Ramallah, said.
"We might see some Fatah officials but, of course, none of the Hamas people because the Palestinian Authority and the president consider that Hamas has carried out a coup."
West Bank violence
"Hamas must decide what their objective is - is it power, is it Israel's destruction, or is it peace for the Palestinian people?
Chris, Stockholm, Sweden
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On Saturday in Ramallah, 50 fighters loyal to Fatah and 200 activists stormed a Palestinian parliament building, dragging a Hamas-backed deputy speaker from the building.
Fatah members also destroyed an Islamic school's offices and local television and radio in Nablus in apparent revenge attacks.
Sami Abu Zuhri, a Hamas official, said 150 Hamas supporters had been "abducted" in West Bank.
He said: "What is happening in the West Bank is a real coup and real terrorism.
"We will not stand handcuffed against these crimes in the West Bank."
Hanan Ashrawi, a Palestinian Legislative Council member, told Al Jazeera that "Palestinian leaders, including from al-Aqsa Martyrs' Brigades, have called on people not carry out acts of revenge in the West Bank".
She, however, said the looting of buildings such as Yasser Arafat's were "acts of extreme provocation".
In Gaza, Hamas fighters ransacked Abbas's bullet-scarred seafront presidential compound and the home of Arafat, former president and Fatah leader, on Saturday.
Many Fatah supporters who live in Gaza fear reprisals from Hamas fighters.
|Haniya has refused to accept his |
dismissal by Abbas [AFP]
Hamas fighters are still searching for their Fatah rivals to seize their weapons and at least 200 Fatah men have already fled the territory, by land or sea to neighbouring Egypt.
Israeli troops at the Erez checkpoint between Gaza and Israel fired warning shots at Palestinians who tried to cross into Israel to escape the violence.
Ismail Haniya, the former prime minister who has refused to accept his firing, has ruled out setting up a separate state ruled by Hamas in the Gaza Strip.
"Gaza belongs to all the Palestinian people and not just Hamas," he said in an interview with France's Le Figaro newspaper.