In a late-night TV address, Morsi rejected an army ultimatum that the crisis be resolved by Wednesday.
Morsi said he would not be dictated to and urged protesters to remain peaceful. However, at least 16 people died at one pro-Morsi rally.
The army earlier leaked details of a draft "roadmap" for Egypt's future.
Details of the plan leaked to media outlined new presidential elections, the suspension of the new constitution and the dissolution of parliament.
The army had warned on Monday that it would step in unless a solution was found, giving Morsi 48 hours to find agreement with the opposition.
That ultimatum expires around 16:30 (15:30 BST) on Wednesday.
However, the unrest shows no sign of abating. The health ministry said that 16 people had been killed and 200 injured at a pro-Morsi rally near Cairo University overnight.
Eyewitnesses said that Morsi supporters had come under attack by unidentified men carrying firearms.
In a 45-minute address on state television, Morsi said he accepted the right to peaceful protest, but said respect for constitutional order was the "only guarantee against further bloodshed."
"When there's violence and thuggery I must act," he said.
Morsi said he would "give my life" to defend constitutional legitimacy,
He blamed the unrest on corruption and remnants of the ousted regime of Hosni Mubarak, and called for protesters to respect the rule of law.
Morsi urged the establishment of a committee of reconciliation as well as a charter of ethics for the media, and said he was prepared to meet all groups and individuals as part of a national dialogue process.
But Mohammed Abdelaziz, a leader of the Tamarod (Rebel) opposition campaign, told AFP: "This is a president threatening his own people. We don't consider him the president of Egypt."
There were outbreaks of violence in several parts of the capital on Tuesday, with casualties reported at hospitals in the north, south and center of Cairo.
More clashes were reported across the country as leaders of the ruling Muslim Brotherhood urged their supporters on to the streets.
On Monday, eight people died as activists stormed and ransacked the Muslim Brotherhood's Cairo headquarters.
Thousands of protesters gathered in Tahrir Square in central Cairo on Tuesday afternoon to demand Morsi step down.
But demonstrations that had been jubilant when the army's ultimatum was interpreted as a coup-in-the-making turned increasingly confrontational later in the day.
Morsi earlier met the head of the armed forces, Gen Abdel Fattah al-Sisi, for a second consecutive day. They did not give any details of the talks, which also included Prime Minister Hisham Qandil.
Morsi was put under further pressure by the resignation of six ministers from his government on Monday, including Foreign Minister Kamel Amr.
On Tuesday, the spokesmen for the presidency and the cabinet were also reported to have quit.
The UN high commissioner for human rights called on the president to engage in a "serious national dialogue" to end the political crisis, and said nothing should be done to undermine the democratic process.
The unrest has escalated since Sunday when Tamarod supporters rallied nationwide, urging the president to step down on the first anniversary of his coming to power.
Morsi became Egypt's first Islamist president on 30 June 2012, after winning an election considered free and fair following the 2011 revolution that toppled Mubarak.
However, his first year in office has been marred by constant political unrest and a sinking economy.