The three injured were members of Hamas's Executive Force.
A Hamas security official accused the Fatah members of trying to plant a bomb near a Hamas security post when their vehicle exploded.
The roadside bombing could stoke tensions between the group and the Fatah faction nearly four months after Hamas seized Gaza by force.
On Wednesday, Hamas ordered the temporary closure of the main Fatah headquarters in the Gaza Strip, citing security reasons.
Hamas said the headquarters was located near where a car exploded on Tuesday night, killing three Fatah members.
Later on Thursday, Israel carried out an air strike in the southern Gaza Strip, an army spokesman said.
The missile missed its target, a car said to be carrying armed men, but injured three civilians nearby, the Islamic Jihad and medical officials said.