The moves against the organization came as an uncertain new political order began to take shape with the swearing in of an interim president as well as the constitution's suspension on Wednesday.
State-run Al-Ahram News reported that Egypt's stock market surged 7% in the first hours of trading Thursday to a near two-month high.
Coup divides Egypt
The coup divided the millions of people who had taken to the streets across Egypt in recent days to defend or criticize Morsy's government.
It also raised questions about what will happen to Morsy and his supporters, who insist he remains the country's legitimate leader; whether violence blamed for the deaths Wednesday of at least 32 people will spread; whether democracy has a chance in Egypt.
But the Tamarod movement that had sought Morsy's ouster was moving on. It said in a tweet that it had nominated Mohamed ElBaradei, an opposition leader, to become prime minister.
ElBaradei told CNN on Thursday that Morsy's ouster was not a coup but was instead a "correction of the uprising of 2011."
Another opposition figure, Egyptian Conference Party leader Amre Moussa, took a similar semantic stance. "This is not a coup; this is a revolution," the former presidential candidate told CNN's Jim Clancy.
Asked whether the Brotherhood arrests were necessary, he said they would be temporary. "There are certain security measures that should not stay but for the first couple of days, three, four days -- the new regime wants to ensure that discipline will take place."
Democratic processes had been "absent" under Morsy, said the former Arab League secretary-general, who lost last year in his bid for the presidency.
Asked whether he would run again, he said, "I have declared several times before that I do not intend to run for president next time. This is my determination as I am talking to you."
The conflicting views, the threat of more violence, possible divisions among the anti-Morsy coalition and Egypt's economic woes represent major obstacles to a smooth transition, said Hani Sabra, director of the Middle Eastern arm of the Eurasia Group, a U.S.-based political risk research and consulting firm. "I don't think that the military's so-called road map is actually going to move smoothly. I think there are a lot of challenges it faces."
The huge crowds that had celebrated Morsy's ouster Wednesday night with horns, cheering, fireworks had thinned hours later. On Thursday, the atmosphere in Cairo's Tahrir Square was calm and celebratory. Crowds cheered as military helicopters flew overhead. Women pushed baby strollers, children had their faces painted, music played and people danced.
Morsy, a Western-educated Islamist elected a year ago, "did not achieve the goals of the people" and failed to meet the generals' demands that he share power with his opposition, Egypt's top military officer, Gen. Abdel-Fatah El-Sisi, said Wednesday in a televised speech to the nation.
Adly Mansour, head of the country's Supreme Constitutional Court, was sworn in Thursday as interim president in Cairo.
At the ceremony, Mansour said the Egyptian people had given him the authority "to amend and correct" the revolution that toppled Hosni Mubarak in 2011.
Until new elections, to be held at an unspecified date, Mansour will have the power to issue constitutional declarations, El-Sisi said.
The Egyptian military has dominated the country for six decades and took direct power for a year and a half after Mubarak's ouster.
Morsy's approval ratings plummeted after his election in June 2012 as his government failed to keep order or revive Egypt's economy.
Morsy's opponents accused him of authoritarianism and forcing through a conservative agenda, and on Monday the military gave him 48 hours to order reforms.
As the deadline neared Wednesday, he offered to form an interim coalition government to oversee parliamentary elections and revise the constitution, which was enacted in January. But those actions failed to satisfy the generals.
The army's move against Morsy and the Muslim Brotherhood, the long-repressed political movement that propelled him to office, provoked wildly conflicting reactions.
In Tahrir Square, the epicenter of two Egyptian upheavals, a vast gathering of Morsy's opponents erupted in jubilation and fireworks at El-Sisi's announcement Wednesday night.