On Friday, Fatah gunmen opened fire on a car carrying Hamas supporters who were calling on people to take part in the rally, a spokesman for the Hamas-led Gaza security force, Islam Shahwan, said.
Earlier, members of the same Hamas-led force shot dead a man from the Fatah-linked al-Aqsa Martyrs Brigades whom they suspected was behind a bombing on Thursday night which killed a Hamas security official and critically wounded another, he said.
More than 30 Palestinians have been killed in fighting between the rival groups since Abbas called last month for presidential and parliamentary elections.
Hamas has said any snap polls would amount to a coup.
Shahwan said five people were detained over Thursday night's bombing, all of them from Fatah. Security forces also tried to arrest Nabil al-Jarjir of al-Aqsa Martyrs Brigades but he was killed after he refused to surrender and instead opened fire.
Fatah accused the Hamas-led security force of "executing" him.
|"Many of the programmes and the goals which the government has started to implement have been hit and obstructed by the outside siege ... and deprived our people of their salaries and food"|
Hamas movement statement
Thursday's bomb blast wounded seven other people, hospital officials said. Three bystanders, including two children, were among those wounded, they said.
Fatah forces also seized a Hamas member and some of his relatives in the northern Gaza town of Beit Lahiya, Shahwan said, triggering more clashes.
Abbas and Khaled Meshaal, the leader of Hamas, pledged on Sunday to curb Palestinian bloodshed after inconclusive talks to form a coalition government which they hope may lift an international boycott imposed because of Hamas's refusal to recognise Israel, renounce violence and abide by interim peace deals.
The Hamas government has also been crippled by Israel withholding Palestinian tax revenues amounting to over $500m.
The Palestinian president also said he expects to hold talks with the United States and Israel within a month on the framework for establishing a Palestinian state.
Abbas said: "I don't have a specific date. Maybe it needs a month, within a month."
Hamas planned to hold a rally in Gaza on Friday to mark a year since its victory over Fatah in Palestinian parliamentary elections.
In a statement issued before the rally, Hamas acknowledged difficulties in the year since its election triumph.
"Many of the programmes and the goals which the government has started to implement have been hit and obstructed by the outside siege ... and deprived our people of their salaries and food," it said.
"It is pitiful to witness the bloody scenes in Gaza between the brothers"
In a veiled attack on Abbas it also criticised "attempts by internal forces to make the government fail".
It said that formation of a unity government remained a "top priority", but it would not abandon "the rights of our people, especially the right of return and the right of resistance".