Security forces forcefully dispersed protesters from a square in central Gaza City late Tuesday, witnesses said.
Protesters in Gaza said security forces set up hundreds of barriers around the main square of demonstrations and were patrolling the area.
They beat people with batons and set fire to tents that were set up by the demonstrators, according to activists in Gaza City.
The March 15 Coalition said hundreds of Hamas security forces stormed the protests and tried to evacuate it by force.
A Palestinian government official, Raed Fattouh, said he was attacked in front of his house, which is near the Katiba square.
The Democratic Front for the Liberation of Palestine condemned the attacks, saying they were directed against peaceful protesters.
In Ramallah, protesters and security forces clashed as the Palestinian Authority sought to gain control of demonstrations calling for an end to rivalry in the occupied territories.
At least 20 people have been injured by security forces in the main square, and six were taken away in ambulances, our correspondent reported from the scene.
The biggest gathering was in Gaza City, where officials from the Hamas-run interior ministry said vast crowds had packed into the city's Square of the Unknown Soldier.
As the protesters demanded that the rival Hamas and Fatah movements patch up their differences, Gaza's Hamas premier Ismail Haniyeh publicly invited his Fatah rival, Mahmoud Abbas, for immediate unity talks in Gaza.
"I invite the president, brother Abu Mazen (Abbas), and Fatah to an immediate meeting here in Gaza ... to start national dialogue in order to achieve reconciliation," Haniya said in a live broadcast after an emergency meeting of his government.
And Abbas proposed holding elections "as soon as possible" in order to end the division.
"I am with the people and in favour of going back to the people to put an end to the divisions through presidential and parliamentary elections," he said after talks in Ramallah with Cypriot President Demetris Christofias.
"No to division!" screamed demonstrators in Gaza City under a sea of red, white, black and green Palestinian flags. "Revolution, revolution until we end the division!"
But the Gaza protests were marred by clashes between dozens of Hamas supporters and hundreds of others in which stones were thrown and three people wounded, witnesses said.
As more and more people packed into the square, disputes could be seen breaking out, an AFP correspondent said, with Fatah cadres locking horns with Hamas activists who were carrying their own green flags and shouting political slogans.
Also, several thousand left the main square to continue their protests at a nearby site to protest Hamas and other political groups co-opting the movement, one of the organisers said.
The rallies, called by the March 15 protest movement, were planned through Facebook by young activists demanding an end to the rift.
Throughout the day, around 7,000 people demonstrated in the West Bank, with around 3,000 in Ramallah, 2,000 in the northern city of Nablus and similar numbers in Hebron, reporters said.
"I'm not from Fatah or from Hamas. I came here with my friends to say enough of this division," said a 24-year-old student demonstrating in Nablus, who gave his name only as Sayed.
"We will stay here until the end of the split," he said.
In Ramallah, demonstrators sang patriotic songs and waved Palestinian flags, but the protest was also plagued with confrontations between supporters of different political groups, marring organisers' attempts to keep the movement apolitical.
One student said there would be no unity unless the politicians decided to give up their entrenched positions.
"We as youth should participate in this event to make a big change, but we also can't stop the division if the politicians don't change," said Sama Musa.
Khalida Jarrar, a Palestinian lawmaker and member of the Popular Front for the Liberation of Palestine, said that "security forces shouldn't be among protesters."
"The youth have a very strong slogan. They want elections for the Palestinian National Council. This is really important, I support this. It's the only thing that can unite Palestinians inside and outside."
The rivalry between Hamas and Fatah dates back to the early 1990s. It soured dramatically after the Islamist movement won elections in 2006 and, a year later, seized control of Gaza after deadly street fighting with Fatah.
Since then, the Palestinian territories have been effectively split in two, with Abbas's rule confined to the West Bank.
In Manara Square, a huge banner was plastered with the faces of dead Palestinians from all the factions, including Yasser Arafat and Hamas spiritual leader Sheikh Ahmad Yassin, emblazoned with the slogan: "End the division!"
The March 15 movement was inspired by a wave of uprisings in the Arab world that has brought down the regimes of Egypt and Tunisia and sparked the revolt in Libya and other Arab countries.
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