Police fired tear gas in a street leading to Cairo's Tahrir Square, heart of the anti-Mubarak uprising in 2011, where thousands demanded Mr Mursi quit and accused him of launching a "coup". There were violent protests in Alexandria, Port Said and Suez.
"The people want to bring down the regime," shouted protesters in Tahrir, echoing a cry used in the uprising that forced Mubarak to step down. "Get out, Mursi," they chanted.
Mr Mursi's aides said the presidential decree was to speed up a protracted transition that has been hindered by legal obstacles, but the president's rivals were quick to condemn him as a new autocratic pharaoh who wanted to impose his Islamist vision on Egypt.
"I am for all Egyptians. I will not be biased against any son of Egypt," Mr Mursi said on a stage outside the presidential palace, saying he was working for social and economic stability and the rotation of power.
"Opposition in Egypt does not worry me, but it has to be real and strong," he said, seeking to placate his critics and telling Egyptians he was committed to the revolution. "Go forward, always forward ... to a new Egypt," he said.
Buoyed by accolades from around the world for mediating a truce between Hamas and Israel, Mr Mursi ordered an Islamist-dominated assembly writing the new constitution could not be dissolved by legal challenges.
"Mursi a 'temporary' dictator," was the headline in the independent daily Al Masry Al Youm.
The president, an Islamist whose roots are in the Muslim Brotherhood, also gave himself sweeping powers that allowed him to sack the unpopular general prosecutor and opened the door for a retrial for Mubarak and his aides.
His decree aimed to end the logjam and push Egypt, the Arab world's most populous nation, more quickly on its democratic path, the president's spokesman said.
"President Mursi said we must go out of the bottleneck without breaking the bottle," Yasser Ali said.
The president's decree said any decrees he issued while no parliament sat could not be challenged, a move that consolidates his power but looks set to polarise Egypt further, threatening more turbulence in a nation at the heart of the Arab Spring.
The turmoil has weighed heavily on Egypt's faltering economy, which was thrown a lifeline this week when a preliminary deal was reached with the International Monetary Fund for a $4.8 billion (£3bn) loan. But it also means unpopular economic measures.
In Alexandria, north of Cairo, protesters ransacked an office of the Brotherhood's political party, burning books and chairs in the street. Supporters of Mr Mursi and opponents clashed elsewhere in the city, leaving 12 injured.
A party building was also attacked by stone-throwing protesters in Port Said, and demonstrators in Suez threw petrol bombs that burned banners outside the party building.
Mr Mursi's decree is bound to worry Western allies, particularly the US, a generous benefactor to Egypt's army. Washington heaped praise on Egypt for its part in bringing Israelis and Palestinians to a ceasefire on Wednesday.
"We are very concerned about the possible huge ramifications of this declaration on human rights and the rule of law in Egypt," Rupert Colville, spokesman for the UN Human Rights Commissioner Navi Pillay, said at the United Nations in Geneva.
The US has been concerned about the fate of a country that was once a close ally under President Mubarak, who preserved Egypt's 1979 peace treaty with Israel. The Gaza deal has reassured Washington, but the deepening polarisation of the nation will be a worry.
Leading liberal politician Mohamed ElBaradei, who joined other politicians on Thursday night to demand the decree be withdrawn, wrote on his Twitter account that Mr Mursi had "usurped all state powers and appointed himself Egypt's new pharaoh".