Addressing the nation in a live televised speech last night, Mursi refused to withdraw the controversial edict he issued and vowed to go ahead with a referendum on the new constitution on December 15.
Mursi said he respects peaceful opposition to his decisions but will not tolerate violence.
He condemned those involved in the clashes -- referring specifically to those with weapons and who are backed by members of the "corrupt ... Ex-regime" -- and promised they'd be held accountable.
"(They) will not escape punishment," the president said.
In his speech, Mursi said more than 80 people had been arrested after days of violent protests.
The struggle between anti and pro-Mursi demonstrators transformed the area around the Presidential Palace in Cairo into a war zone. Tanks and armoured cars were positioned outside the Presidential Palace to keep the protesters at bay.
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He offered to hold dialogue with the opposition and to meet their representatives tomorrow in his office.
Minutes after the speech ended, the Muslim Brotherhood headquarters in Cairo was set "ablaze," state TV reported, citing witnesses.
Police also fired tear gas to disperse hundreds of protesters gathered outside the president's house in his hometown of Zagazig, north of Cairo.
The Islamist group said on its website and Twitter that the building had come under "a terrorist attack," with hundreds surrounding it -- though there was no sign of a fire or significant damage.
On Twitter, the Brotherhood has said it will hold opposition figures "fully responsible for escalation of violence and inciting their supporters." (MORE)