Palestinians had hoped a unity government formed on Saturday between Hamas and Fatah would end months of infighting.
But clashes broke out on Wednesday and spread after a series of abductions.
"The new government should show signs of maturity and responsibility"
Husky, Ottawa, Canada
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The most high profile hostage was Hamdan Assufi, a local Hamas leader whom Hamas has said it would take "measures" to free.
Fatah has not said whether it is holding him. Haniya told Reuters the upsurge in violence was "unfortunate".
An aide said the prime minister had spoken to Abbas on the telephone about the violence and dispatched Hani al-Qawasmi, the new interior minister, to meet Hamas and Fatah leaders in Gaza to "work intensively to put an end to the clashes immediately".
In further violence on Thursday, the house of a Hamas supporter in the northern Gaza Strip was set ablaze.
Hani al-Qawasmi, the interior minister, has
been told to work to end the violence [AFP]
Hamas blamed Fatah and sent members of its executive force to the scene, where a gun battle erupted.
One Fatah activist was injured, local residents and hospital officials said. In an apparent response, the furniture shop of a Fatah supporter was set on fire.
Relatives of a 45-year-old Gazan killed in a separate shooting incident also blamed Hamas executive force members for the shooting.
In response, members of the dead man's clan, one of Gaza's biggest and best armed, torched two executive force vehicles and abducted six members.
Islam Shahwan, a spokesman for the executive force, denied its involvement in the killing of the man and said the death was a result of a feud between two rival families.
In a separate incident in southern Gaza, an officer in a Fatah-dominated security force was shot by unknown attackers.