The deaths were the first from factional fighting in Gaza since November 11. Around 30 other people were also reported injured.
After Hamas took control of the territory after routing Fatah forces in June, Abbas dissolved the national-unity government and named his own, which in effect rules only the West Bank.
Fireworks and firing
Clashes with Hamas security forces erupted in the southern city of Khan Younis after Fatah supporters demonstrated in the street, launching fireworks and firing rifles into the air.
Hamas has banned demonstrations by Fatah supporters.
Ihab al-Ghusein, spokesman for the interior ministry of Gaza-based Hamas government, said that a police patrol came under fire by Fatah elements during attempts to disperse the demonstration.
Medics said three people were killed there, identifying them as two Hamas police officers and a 14-year-old Fatah supporter.
|"I renew the option of early elections ... and I pledge that I will do my best to ensure this election will be the product of a deep and brotherly understanding"|
Mahmoud Abbas, the Palestinian president, on the 43rd anniversary of his party
Officials of Fatah said its supporter was shot by Hamas security forces. Hamas said it was Fatah gunfire that killed him.
Two Fatah men were killed in Gaza City in a clash near a Fatah leader's house, and an elderly man was wounded in a gun battle in northern Gaza and died in hospital Tuesday, according to Dr Moaiya Hassanain of the Gaza health ministry.
Reuters news agency reported the man to be a Hamas member.
Fatah said dozens of its activists were arrested overnight, and that one of its leaders in Gaza City was briefly detained by Hamas forces who shaved off half his hair and mustache as humiliation.
Hamas denied the incident. Islam Shahwan, a spokesman for Hamas security forces, confirmed only that his men made a number of arrests.
Against this backdrop of factional tensions, Abbas urged Hamas to agree to early elections and to cede control of the Gaza Strip.
"I renew the option of early elections ... and I pledge that I will do my best to ensure this election will be the product of a deep and brotherly understanding," he said in the speech marking the Fatah anniversary," he said on Monday.
"I urge all, Fatah and Hamas movements and all other Palestinian factions, to study this alternative and not to rush, as usual, to reject it."
Hamas, which won a Palestinian parliamentary vote in 2006, opposes holding elections before they are due in 2010, citing that any polls prior to this date would be unconstitutional.
Abbas directed some of his comments at Israel.
He called on the Israeli government to cease settlement construction and dismantle its separation barrier in the West Bank.
Abbas said the Israelis "undermine the basis of the independent Palestinian state and block the two-state solution".
He also complained about Israeli military checkpoints in the West Bank, saying: "There are 640 Israeli checkpoints and roadblocks in the West Bank and Israel did not lift or remove any of them."