Many of Gaza 's beleaguered 1.5 million inhabitants called the conflict a civil war and Israel served notice their chances of achieving statehood could dim if Hamas emerged victorious over Abbas' Fatah faction, its partner in a unity government.
Chanting "stop the killing", some 1,000 Palestinians marched through Gaza City, only to draw gunfire that killed two of the demonstrators and wounded four. It was not immediately clear who shot at them.
At least 27 people were killed in the latest violence in which Hamas expanded its attacks to the central and southern Gaza Strip and blew up a Fatah security headquarters, seizing its compound.
The day's carnage raised the death toll since the current surge of bloodshed began on Saturday to 75, hospital officials said.
Palestinians protest violence in Rafah (Photo: AFP)
Hamas' armed wing, which had tightened its hold on northern Gaza by seizing a major Fatah security base and control of main roads, said "the coup-seekers" -- a reference to Fatah -- in that area have until Friday evening to hand over their weapons.
Hours later, a Hamas spokesman offered a conditional ceasefire under which the interior minister, a post now held by Prime Minister Ismail Haniyeh, a Hamas leader, would command all of the Palestinian security services.
"(Egyptian mediators) received the proposal and promised to present it to Abbas. The ball is now in (his) court. Hamas does not set impossible conditions, and if there are serious intentions to resolve the crisis, we will be ready to reciprocate," said Sami Abu Zuhri, a Hamas spokesman.
Gaining groundHamas appeared to have the upper hand in the Gaza Strip, its main stronghold, as fighting spread south towards the Egyptian border. Eighty wounded were taken to hospitals. Gun battles also erupted in Gaza City.
Hamas gunmen killed six Fatah men in one clash, members of Abbas' faction said. Five other Fatah members and a Hamas fighter died in battles in the southern town of Khan Younis, medical officials said.
Two Palestinian employees of the United Nations Relief and Works Agency were killed in separate incidents, and UNRWA said it would temporarily suspend most of its operations in the coastal territory.
In Washington, White House spokesman Tony Snow said the Palestinians "are going to have to sort out their politics and figure out which pathway they want to pursue, the pathway towards two states living peaceably side by side or whether this sort of chaos is going to become a problem."
Israeli Foreign Minister Tzipi Livni, speaking in Jerusalem, said Hamas control of Gaza would raise questions about Israel's "ability to reach agreements with (Abbas) and whether it would be possible to implement them" in the territory.
Addressing reporters in the West Bank, an area largely untouched by internal fighting, Abbas said, "what is happening in Gaza is madness" and repeated a call for a truce.
The Gaza bloodshed has prompted Fatah to say it was suspending participation in the unity government with Hamas without an immediate ceasefire. The government was formed in March under Saudi mediation to try to end infighting and ease Western sanctions.
Abbas' group stopped short of withdrawing outright, a move that could lead to presidential rule by decree and widen a divide between the West Bank, where Fatah is dominant, and Gaza .