Before the clashes broke out, Hamas had accused Fatah of ordering Thursday night's attack on Ismail Haniya, the Palestinian prime minister, in which one bodyguard was killed.
Hospital sources said at least 20 people were wounded in the West Bank shooting, some of them critically.
Al Jazeera's West Bank correspondent said the Hamas supporters had been marching to Ramallah's central square to attend the festivities. The security forces used clubs and rifles to beat them back before shooting broke out.
In a show of force, Hamas deployed hundreds of heavily armed men across Gaza City and called on Abbas to remove his presidential guard, Force 17, from the streets.
Hamas had earlier accused the guard of involvement in the attack on Haniya and had called for it to be taken off the streets.
In the Gaza Strip, the rival groups fired at each other on the streets. It was unclear if anyone was hurt.
The body of Haniya's bodyguard was carried through the streets before being buried on Friday.
Another bodyguard, the prime minister's son and a political adviser were wounded in Thursday night's incident, which Hamas says was an assassination attempt.
Accusation denied
Ismail Rudwan, a Hamas spokesman, singled out Mohammed Dahlan, a Fatah politician, as being behind the armed attack on Haniya's convoy late on Thursday as it was leaving the Rafah border crossing between Gaza and Egypt.
Dahlan said that the accusations "are not worth answering".
He blamed Hamas "gangs" for the violence, which has surged since unidentified militants shot dead three young sons of an intelligence official loyal to Abbas outside the boys' Gaza school early this week.
Abdel-Hakim Awad, a spokesman for Fatah in Gaza, said: "It [Hamas] is pouring oil on the fire ... and bears full responsibility for the escalation that may result."
Israel had blocked Haniya from entering Gaza until he agreed to leave behind $35m in cash raised during a fundraising tour that included Iran and Syria.