Waving yellow Fatah flags, supporters from across the territory gathered in Gaza City to celebrate the movement founded by the late Yasser Arafat.
Some held photographs of the two former and current political leaders.
West Bank-based Fatah cancelled its anniversary event last year after dozens of its supporters were arrested by Hamas authorities.
The interior ministry in Gaza granted permission for Wednesday's rally to go ahead, according to a Hamas leader.
"We consented to holding these festivities on Al-Wehda Street for Fatah to champion the unity of the Palestinian people," he told AFP, requesting anonymity.
"It's a referendum on the public support for Fatah and the leadership of Abu Mazen," he added, using a common nickname for Abbas.
Islamist movement Hamas seized control of Gaza from Abbas's forces in 2007 and the two sides have remained split, with the Palestinian president based in the occupied West Bank.
No Palestinian elections have been held since 2006, apart from local polls, but Abbas has recently pushed for a vote.
Hamas called for a presidential decree setting the election date, but Abbas said he wants Israel to first guarantee that the vote can take place in East Jerusalem.
"We will not accept elections without Jerusalem and without our people... Jerusalem is the capital of our state and it is not up for sale or negotiation," Abbas said in a speech aired at the rally.
Israel occupied East Jerusalem in 1967 and later annexed it, in a move never recognised by the international community.
The Palestinians see East Jerusalem as the capital of their future state, but Israel has declared the whole city its undivided capital.
Abbas has previously pledged on multiple occasions to hold elections but without any results.
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