Morsi issued constitutional amendments on Thursday, giving himself powers above any court as the guardian of Egypt's revolution. At the same time, he ordered the retrial of leaders of ousted president Hosni Mubarak's regime for the killing of protesters in last year's uprising.
In Cairo, thousands of protesters converged on Tahrir Square following Friday midday prayers, led by prominent pro-democracy figures like Mohamed ElBaradei, former head of the UN's nuclear agency.
ElBaradei called Morsi a "new pharaoh," and others said his new decrees put him in the same category as Mubarak, whose autocratic practices and harsh crackdowns led to the revolt against him.
Chants of "leave, leave" rang across the square, echoing calls against Mubarak at the epicentre of last year's uprising, as crowds of protesters circled the square chanting "Morsi is Mubarak" and "revolution everywhere."
Police fired tear gas at the edge of the square as some protesters threw rocks. Other anti-Morsi demonstrators lit small fires in a street leading to the square.
State TV reported that in the Suez Canal cities of Port Said, Ismailia and Suez, the offices of Freedom and Justice, the political arm of the Muslim Brotherhood, had been torched.
Violent clashes also erupted between thousands of pro-democracy protesters and Islamists in the Mediterranean city of Alexandria. Anti-Morsi protesters there stormed the office of the Muslim Brotherhood.
At a rally outside the presidential palace, Morsi told supporters that Egypt was on the path to "freedom and democracy" and that he wants the country to have a "real and strong opposition" that operates in a "clear and clean way."
"Political stability, social stability and economic stability are what I want, and that is what I am working for," he said, adding everyone in the country has equal rights. "We'd never be able to side with one party against the other."
One of his decrees exempts the country's Islamist-dominated constituent assembly, which is writing Egypt's new constitution, from judicial review. Another gives Morsi the power to take "due measures and steps" to deal with any "threat" to the revolution, national unity and safety or anything that obstructs the work of state institutions.
"Morsi is essentially taking a bit of a gamble here. On the one hand he has announced to the people in the square that he is going to retry security officials and others from the old regime who were acquitted of killing protesters during the revolution and in the months that followed. That's something people on the street here have been demanding for a long time," CBC's Margaret Evans reported.
"On the other hand, he has built these protective measures around himself. People say he's trying to grab power for himself and that they fear he's trying to steal the revolution," Evans said.
Morsi has also fired the country's top prosecutor, Abdel-Maguid Mahmoud — a Mubarak-era appointee — replacing him with Talaat Abdullah, a career judge.
Rights groups have likened Morsi's declarations to new emergency laws.
Among those calling for protests is former presidential candidate Sameh Ashour, who on Friday accused Morsi of being a worse dictator than Mubarak.
Rupert Colville, a spokesman for UN Human Rights Commissioner Navi Pillay, said Morsi's announcement raises concerns over human rights.
"We are very concerned about the possible huge ramifications of this declaration on human rights and the rule of law in Egypt," Colville told reporters in Geneva. "We also fear this could lead to a very volatile situation over the next few days."
Morsi's power base consists mostly of backers of the Islamist Muslim Brotherhood. He took office last June after winning 52 per cent of the vote to become Egypt's first freely elected and civilian president, ending nearly six decades of de facto military rule.