Hamas said three members of its security forces were also hurt on Tuesday when Fatah elements fired a rocket-propelled grenade at their vehicle.
Fatah denied any role in the shootings, accusing Hamas of stockpiling weapons.
Abdel-Hakim Awad, a Fatah spokesman in Gaza, told a news conference: "Hamas does not believe in national unity except as a slogan. Hamas's policy is based on excluding others and wiping out others."
Hamas has vowed to double its security force strength to 12,000 members, but Awad said that it should be disbanded.
Factional fighting has worsened since Mahmoud Abbas, the Palestinian president, called last month for early elections in his power struggle with the governing Hamas faction.
Tensions rose further at the weekend after Abbas declared Hamas's "Executive Force" illegal.
Also on Tuesday, tribal dignitaries and representatives of Palestinian factions in Hebron held a conference at which they urged people to deem the spilling of Palestinian blood a taboo, Al Jazeera said.
In a similar appeal, Sheikh Hassan Yusuf, a senior Hamas leader, and Brigadier Ahmed Mahmoud Dhamra, commander of the Palestinian presidential guards, who are both detained in Israel, asked Fatah and Hamas to exercise restraint.
They called for a halt to hostile media campaigns and emphasised the need for Palestinian national unity, Al Jazeera said.