Hamas seized the coastal territory after routing Fatah forces in June, and has since banned public gatherings by Fatah supporters.
Violence erupted when Fatah supporters defied the Hamas ban and demonstrated throughout the Gaza Strip.
Witnesses said gunfire echoed throughout the streets of Gaza, and Hamas declared a full alert among its forces.
Mahmoud Abbas, the Palestinian president, has meanwhile urged Hamas to agree to early elections and to open a "new page" by ceding 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," Abbas said in a speech marking the Fatah anniversary.
"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.