"It is my duty ... to protect institutions of the nation," he said. "I will always fulfill this role, no matter how much pressure or what the situation."
Opposition protesters jeered and raised their shoes in contempt.
"We have two simple demands: Cancel the decrees and change the draft constitution. Other than that he can just go away," shouted one protester, Osama El-Sayyed.
"I have no hope in this man" shouted another as thousands chanted "Erhal! Erhal!" — "Leave! Leave!" in Arabic.
Later, a photograph of Morsi giving his speech was circulated on social networking sites alongside one of Mubarak addressing the nation during the 18-day uprising that toppled his 29-year rule in February 2011. Both wore black ties and dark suits.
The opposition issued a statement rejecting Morsi's offer of a dialogue, and spokesman Hussein Abdel-Ghani dismissed Morsi's address.
"Tonight, he proved that he is not a president for all Egyptians, but merely the representative of the Muslim Brotherhood in the presidency," Abdel-Ghani said on state television.
Earlier Thursday, the Egyptian army and elite Republican Guard sealed off the presidential palace with tanks and barbed wire, following the worst night of violence of the two-week crisis.
Responding to a call to "protect" the presidential palace, thousands of Brotherhood members and other Islamists descended on the area Wednesday, beating and chasing away some 300 opposition protesters who had been staging a peaceful sit-in there. Hours of street battles followed.
"We raise Egypt's flag but they raise the Brotherhood flag. This is the difference," Cairo protester Magdi Farag said as he held the tri-colored national flag stained with blood from his friend's injury in the clashes.