The truce took effect after Ismael Haniya, prime minister of the Hamas-led government, met an aide to Abbas, the leader of the Fatah movement, on Monday to try to stem a surge of fighting.
Hamas fighters removed their impromptu checkpoints, but some Fatah fighters remained visible in Gaza City, protecting the official residences of Abbas and a senior faction leader as well as the compound of a security service considered loyal to Fatah.
The ceasefire also requires Hamas and Fatah fighters to release hostages and for Palestinian police to deploy in force.
There was a limited police presence in Gaza on Tuesday, but no immediate word on the release of hostages.
Abbas, in Cairo, meeting the Egyptian president Hosni Mubarak, condemned the attack in the Israeli Red Sea resort of Eilat on Monday that killed three people and was carried out by a suicide bomber from Gaza.
“My position regarding this operation is that I do not accept it and I reject and condemn it,” Abbas said.
“There was no need at all, and it does not benefit us at all. I do not think that this operation in particular will impact the calm between us and the Israelis in the Gaza Strip.”
Meanwhile Amir Peretz, the Israeli defence minister, vowed to take action but gave no hint as to when, where or against whom the military would strike.
“The initiative will be ours and we have no intention of relaying what we plan to do,” Peretz said.
With the Quartet of Middle East peace mediators due to meet in Washington on Friday, Israel may be wary of taking military action that might jeopardise a diplomatic drive promoted by its main ally, the United States.
In remarks on Monday after the Eilat attack, Peretz said Israel would “do everything to preserve” a two-month-old Gaza ceasefire with fighter groups. The suicide bombing was the first in Israel in nine months.
Speaking after Haniya and Abbas met, Mahmoud al-Zahar, the Palestinian foreign minister, said the two sides agreed to hold their fire and remove all their fighters from the streets of Gaza.
In the run-up to the truce, at least 30 Palestinians were killed in five days of factional street battles.
They were the fiercest clashes since Hamas trounced Fatah in an election last year, triggering a Western aid embargo.
The violence has derailed unity government talks between Hamas and Fatah. Many shops and schools have been closed.
Previous ceasefires, including one last month, have been short-lived.