Mursi, who was speaking on the eve of his swearing-in, took the symbolic oath after many protesters called on him to do so to defy the ruling army who took power after Hosni Mubarak's ouster.
He swore to uphold the constitution and "the republican system", reciting the words of an oath which he will formally take tomorrow in front of the supreme constitutional court.
Mursi, 60, in his first public speech promised to be a "president for all Egyptians", adding: "You are the source of all authority and legitimacy."
He insisted that "no institution will be above the people," critiquing an army which has sought to shield itself from parliamentary oversight.
"I promise you that I will not give up on any of the powers given to the president," Mursi said, in a veiled reference to the Supreme Council of Armed Forces' recent decrees.
At one point he opened his jacket to show the crowd he wasn't wearing a bulletproof vest, saying he "fears no one but God." His defiant speech was a clear challenge to the army, which also says it represents the will of the people.
"We will complete the journey in a civil state, a nationalist state, a constitutional state, a modern state," he told the crowd, to applause and cheers.
Mursi promised to end torture and discrimination, and to deliver social justice for millions of Egyptians.
He chanted with the crowd at the beginning of his speech, "Free revolutionaries will continue the course."
"Commissioning me with the presidency is a great honour I'm proud of," Morsy said.
A wave of emotion swept the Egyptian capital as thousands thronged the iconic Tahrir Square to hear Mursi as a show of angst against the ruling military and a show of support to Mursi, who as president elect is negotiating with the Army the powers his civilian administration will wield.
Mursi also praised crowds in Cairo's Tahrir Square, the focal point of protests that overthrew Mubarak.
"The revolution must continue until all is objectives are met," Mursi said. (more)