A planned march by those supporting Morsi was canceled to avoid clashes with anti-Morsi protesters, although pro-Morsi rallies were held in Alexandria and other cities, Ahram Online reported.
An anti-Morsi march at Al-Azhar University in Cairo was attacked by members of the Muslim Brotherhood, to which Morsi belongs.
An estimated 500 attorneys chanting "Egypt is a state of law" demonstrated in front of their bar association building. Groups of protesting journalists and students gathered at various points around the capital.
In the Upper Egyptian governorate of Bani Sueif, protesters chanted "down down with Brotherhood guide rule" and "take off your beard and you will find that underneath it is Mubarak's face."
The demonstrations were organized in support of protesters in Cairo's Tahrir Square, where more than 1,000 people were reported to have gathered. They were joined there by thousands of protesters from the Nile Delta.
South of Cairo, in Assiut governorate, thousands of Al Aztar students and faculty members supporting Morsi marched to the governorate's headquarters calling for a "purging" of the judiciary.
Morsi's decree, issued Thursday, simply asserts presidential powers already approved by the courts, presidential spokesman Yasser Ali said on Egyptian television after Morsi met with the judges of the Supreme Council of the Judiciary, the highest council overseeing the Egyptian courts.
Morsi did not seek to put himself above judicial oversight, Ali said. On the contrary, Morsi had "the utmost respect for the judicial authority and its members and takes into account its immunity and specialties," he said.
Ali said after the 90-minute meeting that the judges were "understanding" of the reasons for Morsi's action.
The Supreme Judicial Council met at noon to discuss its Monday night meeting with Morsi. Members said they were still against the decision.
Some 444 people have been injured in protests following Morsi's declaration, Bikya Masr reported.
Two people have died. A third person was reported to have died Monday night from injuries received in a Sunday protest.