"peaceful" protests in a statement early on Saturday, demanding the military restore the country's first democratically elected leader.
In the restive north of the Sinai peninsula, armed Morsi supporters stormed the provincial headquarters in the town of El-Arish after a gunfight and raised the black banner of Al-Qaeda-inspired Islamist militants, an AFP correspondent said.
At least 12 people were killed in the Mediterranean city of Alexandria as Morsi's supporters and opponents fought a pitched street battle, the official MENA news agency said.
Police continued to round up top Islamists, announcing the arrest of Khairat al-Shater, widely seen as the most powerful man behind Morsi in the Muslim Brotherhood movement.
A spokesman for UN chief Ban Ki-moon quoted him calling for a peaceful end to the crisis. "There is no place for retribution or for the exclusion of any major party or community".
In Cairo's iconic Tahrir Square, at least two people were killed when Morsi supporters traded fire with his opponents, state television reported. The clashes subsided when the army separated the protesters using armoured vehicles.
"We are not taking sides. Our mission is to secure the lives of protesters," military spokesman Colonel Ahmed Ali told AFP.
Four protesters were killed outside the Republican Guard headquarters after breaking away from a pro-Morsi demonstration, the official MENA news agency reported.
The bodies of two people were covered with sheets, said an AFP correspondent, adding that another protester was shot in the head. Soldiers had warned a protester waving a picture of the ousted president not to approach their barbed wire cordon.
They opened fire when he ignored them, and shots were then heard from both sides, an AFP reporter said. The Islamists accuse the military of conducting a brazen coup against Morsi, Egypt's first democratically elected president, after millions called for his ouster on the June 30 anniversary of his first turbulent year in power
Friday's violence came as the supreme guide of Morsi's Muslim Brotherhood, Mohammed Badie, vowed that members of the Islamist movement would throng the streets in their millions until his presidency is restored.
Badie appeared at Cairo's Rabaa al-Adawiya mosque to screams of joy from jubilant supporters, following reports he had been detained after Wednesday's ouster of the president.
"Millions will remain in the squares until we carry our elected president, Mohamed Morsi, on our shoulders," Badie told the crowd, before leading chants of "Military coup!" and "Invalid!"
Violence between Morsi's supporters and opponents also left one protester dead at Assiut in central Egypt and another in Minya, officials said. In the Sinai, where gunmen killed five policemen and Islamists killed a soldier in a machinegun and rocket attack.
In El-Arish, at least 16 people were wounded in clashes before armed Morsi supporters stormed the provincial HQ. The armed forces have already sworn in Adly Mansour as interim president, and he issued his first decree on Friday, dissolving the Islamist-led parliament and appointing a new intelligence chief.
Before Friday's rallies, around a dozen low-flying military jets screamed across Cairo, but the show of force failed to deter Morsi's supporters. Morsi, who has not been seen since Wednesday, had issued a defiant call for supporters to protect his elected "legitimacy", in a recorded speech aired hours after his removal.
The military had said it supported the right to peaceful protest, but warned against violence and acts of civil disobedience. Ahead of Friday's rallies, Mansour had called in a television interview for unity. "All I can say to the Egyptian people is to be one body. We had enough of division," he told Britain's Channel 4.
Prominent liberal leader Mohamed ElBaradei defended the military's intervention, saying "the other option was a civil war. "We were between a rock and a hard place, and people need to understand that," the former UN nuclear watchdog chief told the BBC. Army chief General Abdel Fattah al-Sisi announced Morsi's overthrow on Wednesday night, citing his inability to end a deepening political crisis.
Military police rounded up senior Brotherhood members, although two were later released. Morsi himself was "preventively detained", a senior officer told AFP. A judicial source said the prosecution would on Monday begin questioning Brotherhood members, including Morsi, for "insulting the judiciary".
Morsi's rule was marked by accusations that he concentrated power in the hands of the Brotherhood. His supporters argue Morsi was confronted at every turn with a hostile bureaucracy left over by Hosni Mubarak, overthrown in the Arab Spring-inspired uprising of 2011.