Hamas prevented voting in the Gaza Strip, the territory the group seized from Abbas in 2007, and boycotted the contest in the West Bank. Hamas argues that elections can only be held once Hamas and Fatah reconcile.
"We ask to stop this disgrace," said Hamas spokesman Fawzi Barhoum, dismissing Saturday's vote as meaningless.
Saeb Erekat, a senior Abbas aide, countered that "Hamas cannot have a veto on democracy." Critics say the group banned voting in Gaza to prevent largely vanquished rivals, particularly from Fatah, from gaining a new foothold there.
Fatah, though running virtually unopposed in the West Bank, could be bruised by low turnout or a victory of party renegades competing against Abbas-endorsed candidates in several of the larger towns.
The election was overshadowed by widespread voter apathy and a general sense of malaise.
Abbas' Palestinian Authority, a self-rule government in parts of the Israeli-controlled West Bank is facing a slew of difficulties: it is mired in a chronic cash crisis; efforts to heal the Palestinian political split have failed; prospects are virtually nil for resuming meaningful negotiations with Israel's conservative government on setting up a Palestinian state in the West Bank, Gaza and east Jerusalem, the territories Israel captured in 1967.
"People are crushed by heavy burdens," said Mohammed Nasser, 25, an accountant in the city of Ramallah who planned to stay at home. "Would these elections solve our problems? Of course not."
Others did vote, including Amani Qasim, 30, who said she wanted to see new faces in Ramallah's city council. By midday, turnout was 24 percent, election officials said.
About 515,000 voters were eligible to choose new councils in 93 towns and villages in the West Bank, picking from lists of candidates rather than individuals. In an additional 179 communities, residents reached power-sharing deals, many brokered by clan leaders, and decided to forgo elections.
In 82 other villages, there were no candidates, said election official Fareed Tomallah.
Polls opened at 7 a.m. and closed 12 hours later. One polling station, near the city of Nablus, stayed open for an additional hour to accommodate the Samaritans, followers of an offshoot of Judaism, who observed the Sabbath on Saturday and couldn't vote until after nightfall.
Turnout was above 50 percent, according to an initial estimate by election officials.
Hamas asked its activists and sympathizers to stay home. "Our supporters understand that we are not participating, and therefore we expect them not to vote for anyone," said Ahmed Atoun, a Hamas lawmaker in the West Bank.