At least 18 people were reported injured in fighting in Tahrir Square in Cairo, where many of the protests that led to the overthrow of Hosni Mubarak were held, Ahram Online reported. Members of the April 6 Youth Movement and the Constitutional Party also held a demonstration in Sharm-el-Sheik, a resort on the Red Sea.
Morsi, formerly a leader in the Muslim Brotherhood, addressed supporters at the presidential palace, promising to protect the country's stability and the rights of his opponents, the BBC reported.
"I am for all Egyptians. I will not be biased against any son of Egypt," he said.
Abdel-Meguid Mahmoud, a former prosecutor general, promised a legal challenge to Morsi's constitutional declaration, Ahram reported.
On Thursday, Morsi issued a decree stating that all decisions he has made since taking office are not subject to review or appeal.
Muslim Brotherhood offices were seen on fire in the Suez Canal cities of Suez, Port Said, and Ismailia, Voice of America reported.
Meanwhile, thousands of Morsi's backers gathered outside of the presidential palace, the BBC reported.
Despite the protests Morsi has vowed to protect the rights of the opposition, saying, "There is a great future for this nation."
Mohamed ElBaradei, Nobel Peace Prize laureate who heads the lawyers syndicate and is a key opposition figure, sent out the call for Egyptians to protest Friday, BBC reported.
The president "appointed himself Egypt's new pharaoh. A major blow to the revolution that could have dire consequences," ElBaradei wrote on Twitter.
Protesters have picked up on the moniker and are referring to Morsi as the "new pharaoh," Ahram Online reported.
Security forces and protesters clashed throughout Thursday night into Friday morning along Mohamed Mahmoud Street in Cairo, near Tahrir Square.
"We have had around 20 major injuries last night, all from tear gas," said paramedic Mohamed Hamdi.