"There is no parliament and there is no constitution," said Hamdi Nayim, who joined the celebration in the square. "We will not be satisfied if the army will control us and govern us here."
Each side in the election accused the other of voting irregularities and called for an investigation.
Shafik's campaign filed more than 100 complaints, alleging "ballot rigging and stuffing."
It accused the Muslim Brotherhood of having bribed voters with "large sums of money and food" to back Morsi, while intimidating and threatening violence against Shafik's supporters.
The Muslim Brotherhood, in a statement posted on its website, denied the allegations and accused Shafik's camp of bribing voters.
But longtime Egyptian journalist Hani Shukrallah said that the most significant change is the one that has affected the minds of the voters.
"They have a sense of their own rights, they have a sense of their personal dignity, they are convinced that they can," he told CNN. "They look at the state as their servant and not their master, and this is something very new."
From Washington, Pentagon officials were eyeing events in Egypt in hopes of maintaining a relationship with its military. "We'd like it to continue," said Pentagon spokesman Capt. John Kirby.
But Pentagon Press Secretary George Little said U.S. officials expect that SCAF "will transfer full power to a democratically elected civilian government," as it had promised. "We have and will continue to urge the SCAF to relinquish power to civilian elected authorities and to respect the universal rights of the Egyptian people and the rule of law," he told reporters.
The United States spends $2.7 billion per year on military financing, education and training in and to Egypt, Kirby said.