She said that it was "the second week in a row [that] Fatah and other PLO factions" have called for protests at Friday prayers.
The calls went out "in two separate locations in Gaza City alone, while other prayers were held in Khan Younis and in Rafah in the Southern Gaza Strip," she said.
She said that the security forces arrested dozens of the protesters.
The Fatah supporters were protesting against Hamas, which seized control of the the Gaza Strip in June. A similar protest last Friday also ended in clashes.
Six Palestinian youths were wounded by what appeared to be stun grenades as Fatah supporters threw rocks at the home of a leading Hamas figure in the southern town of Rafah, medics said.
Associated Press reports said that Hamas men began firing into the air to disperse the crowd, then began arresting protesters and taking them away in jeeps and beat several demonstrators.
A preacher at the Gaza City rally urged worshippers to put aside factional differences and unite against Israel.
Rushdi al-Zayan, a preacher at the rally, told the crowd.: "Mosques should be independent of armed difference and dispute ... We are one people and our enemy is one. We should unite against our enemy."
Fatah says hundreds of its supporters have been detained in Gaza since the Hamas takeover but most have been freed.
Hamas says over 400 of its followers have been held in the West Bank.