Islamists and their opponents threw stones, bottles and petrol bombs, and some fought hand-to-hand, showing how feelings still run high between the rival groups trying to shape the new Egypt after decades of autocracy, even though the streets have generally been calmer since Morsi's election in June.
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The Health Ministry said 110 people had sustained light to moderate injuries, state media reported.
A government is in place, but Islamists and liberals are at loggerheads over the drafting of the new constitution, which must be agreed before a new parliament can be elected.
Demonstrators in Tahrir Square, Friday (Photo: AP)
Many of the thousands who gathered in Tahrir Square were angry at this week's court ruling that acquitted former officials charged with ordering a camel and horseback charge on protesters in the uprising that ousted Hosni Mubarak last year.
But even before that ruling, Morsi's opponents had called for protests against what they say is his failure to deliver on his promises for his first 100 days in office.
"Down, down with rule by the guide," Morsi's opponents chanted, suggesting that Muslim Brotherhood leader Mohamed Badie pulls the strings even though Morsi officially quit the Brotherhood on taking office.
"Morsi, Morsi," the president's backers responded.
Some demonstrators pulled down a temporary podium that had been erected on one side of the square for speeches. Later, Islamists took over the square, triggering scuffles in nearby streets as they tried to keep rival groups out.
"We went to protest against the constituent assembly and Morsi's failure in his 100 days, and Islamists prevented us and are now controlling the square," said Islam Wagdy, 19, a member of a group set up by leftist politician Hamdeen Sabahy.
A member of the Brotherhood's Freedom and Justice Party dismissed that account. "What happened today was an attempt by the liberal powers ... to prevent Islamists expressing their views and protesting in Tahrir, which belongs to all Egyptians and not to a certain current," said the FJP's Ahmed Sobeih.
There was no intervention by police, who have often been the target of protesters' anger in the past because of their brutality against demonstrators in last year's revolt.
Camel charge acquittals
The Brotherhood, which joined Friday's protest, had put the focus for the demonstration on this week's court ruling.
The charge by men on camels and horseback was one of the most violent incidents of the uprising that ousted Mubarak in February 2011. The case has been closely watched by those seeking justice for the hundreds killed in the revolt.
The court acquitted top Mubarak-era officials such as former lower house speaker Fathi Sorour and Mubarak aide Safwat Sherif, both of whom are detested by many Egyptians.
Demonstrators also gathered in Egypt's second city of Alexandria, where Morsi went to a mosque to perform Friday prayers before giving a speech there.
"We won't let anyone involved in corruption get away," he said, while urging protesters not to disrupt people's work. As he spoke, some chanted: "The people want the judiciary purged."
Many blame the general prosecutor, perceived as a Mubarak loyalist, for not securing convictions.
In an apparent bid to appease the public, the president said late on Thursday he was moving Abdel Maguid Mahmoud out of that position to make him ambassador to the Vatican, because Egyptian law prevented him being dismissed.
Mahmoud denounced the move and told Egyptian media he would stay on. The influential judges' club condemned the decision as interference and called for a meeting of judicial officials on Sunday to discuss action, the state news agency reported.
Even some political groups who wanted Mahmoud out questioned the way Morsi had done it. The liberal Free Egyptians Party said changing the prosecutor should be an independent judicial move.
Morsi has won grudging respect from some opponents for pushing the army out of politics, after decades of rule by military men, and for raising Egypt's profile abroad.
But many Egyptians, with high expectations after the revolt, say he has not done enough at home, failing to deliver on promises for his first 100 days such as cleaning up cities and getting traffic moving in Egypt's congested streets.
Many more secular-minded Egyptians and minority Christians also worry that Morsi and his Islamist supporters will seek to impose religious restrictions on society.