President Mohammed Mursi has appeared before supporters in Cairo to defend a new decree that grants him sweeping powers.
He told them he was leading Egypt on a path to "freedom and democracy" and was the guardian of stability.
He was speaking as thousands of opponents gathered in Cairo's Tahrir Square and offices of the president's party were attacked in several cities.
The decree says presidential decisions cannot be revoked by any authority.
'Mursi is Mubarak'
Speaking at a rally at the presidential palace in Cairo, Mr Mursi said he was working to secure a strong and stable nation, for which there was a "great future".
He said: "I am for all Egyptians. I will not be biased against any son of Egypt."
Mr Mursi said he was the guardian of political, economic and social stability and wanted to see a "genuine opposition, a strong opposition".
"I am the guarantor of that and I will protect for my brothers in the opposition all their rights so they can exercise their role."
Mr Mursi also vowed to defend the independence of the executive, judiciary and legislature and not issue decrees to settle scores.
But across the capital in Tahrir Square, thousands of the president's opponents heeded calls to demonstrate against the decree.
Chants of "Mursi is Mubarak... revolution everywhere" rang out.
There were clashes between protesters and police in the square, with tear gas fired at demonstrators and Molotov cocktails thrown in return.
According to Egypt's state-run news agency, Mena, three people were injured in violence in Cairo's central Mohammed Mahmoud street.
Anti-Mursi protesters from more than 20 different groups have now begun a week-long sit-in at Tahrir Square, and have called for a huge protest on Tuesday.
Offices of the president's Muslim Brotherhood party have reportedly been attacked in the cities of Port Said and Ismailia.
Clashes between rival demonstrations took place in Alexandria.
Protesters in the Mediterranean city stormed the offices of the Muslim Brotherhood's political wing, the Freedom and Justice Party, throwing out books and chairs, and starting a fire.
Up to 2,000 demonstrators stormed the Muslim Brotherhood headquarters in Suez, while hundreds of people also protested against the new decree in the Red Sea resort of Sharm el-Sheikh.
Almost 100 people were injured across the country, the health ministry said.
In a joint news conference on Thursday, Sameh Ashour, head of a lawyers association, and key opposition figures Mohamed ElBaradei and Amr Moussa accused Mr Mursi of "monopolising all three branches of government" and overseeing "the total execution of the independence of the judiciary".
Mr ElBaradei, a Nobel Peace Prize winner, wrote on his Twitter account that the president had "appointed himself Egypt's new pharaoh. A major blow to the revolution that could have dire consequences".
US state department spokeswoman Victoria Nuland said on Friday that the decree had "raised concerns" in the international community, because Egypt's revolution "was to ensure that power would not be overly concentrated in the hands of any one person or institution".
She said the US wanted "democratic dialogue" within Egypt to solve constitutional issues.
The new decree bans challenges to Mr Mursi's laws and decisions, and says no court can dissolve the constituent assembly, which is drawing up a new constitution.
It also opens the way for a retrial of people convicted of killings during Egypt's 2011 uprising which toppled President Hosni Mubarak.
The declaration also gives the 100-member constituent assembly two additional months to draft a new constitution, to replace the one suspended after Mr Mubarak was overthrown.
The rewrite of the constitution, which was meant to be finished by December, has been plagued by lawsuits questioning the make-up of the constituent assembly.
Once completed, the document is due to be put to a referendum. If it is approved, legislative elections will be held two months later.