Both Hamas and Fatah, however, have accepted an invitation from King Abdullah of Saudi Arabia for peace talks to be held at Mecca's Grand Mosque.
The Saudi Press Agency quoted Abdullah as saying: "I invite them all ... to discuss disputes in a neutral [environment] without intervention from any other side."
Residents said some families in Gaza were leaving their homes near the worst of the fighting.
Nour Odeh, Al Jazeera's Gaza correspondent, said: "At the moment, the situation is very tense on the streets of Gaza and some streets have been completely closed off to civilians."
The violence has derailed coalition talks between the two factions and brought much of Gaza to a standstill.
Disturbances also spread to the West Bank city of Nablus, where Fatah fighters kidnapped 11 Hamas members, security sources said.
Fatah supporters later set ablaze the Nablus offices of the Hamas-controlled education ministry and kidnapped five employees, Ahmed Doleh, the director of the ministry, said.
Armed men from Fatah-affiliated al-Aqsa Martyrs' Brigades seized a local Hamas leader in front of local news crews in Nablus.
As the clashes intensified, Ismael Haniya, prime minister of the Hamas government, issued a plea urging an end to the conflict.
"We call on all the Palestinian people to protect national unity, to make the language of dialogue and reason prevail, to withdraw weapons from the streets, and put an end to the tensions," Haniya said.
Laila El-Haddad interviews members of armed groups allied to Hamas and to Fatah to discover why they do what they do - read An eye for an eye in Gaza
An explosion early on Sunday morning rocked the Gaza City home of a bodyguard of Mohammed Dahlan, a powerful Fatah commander. No casualties were reported.