Mansour praised the army and the huge protests that led to Morsi's removal.
He added that Egyptians, who spent 30 years with strongman Hosni Mubarak in the presidency, should maintain their revolutionary ideals of 2011 and not "worship" one leader.
''The guarantee of the continuity of the spirit of the revolution carries the hope for us that the values of this revolution will be upheld, the first of which is to put an end to the idea of worshipping the leader and creating a half-divine entity out him and to end all kinds of impunity that the weak bestow on the rulers and presidents," Mansour said.
He said that Morsi's Muslim Brotherhood was a segment of the Egyptian public and was welcome to help "build the nation."
However, Egypt's state news agency MENA said prosecutors have ordered the arrest of the top two leaders of the Muslim Brotherhood.
MENA said arrest warrants were issued for the group's spiritual leader, Muhammad Badie, allegedly for inciting the killing of peaceful protesters, and for his deputy, Khayrat al-Shater, on the same charges.
On July 3, army commander General Abdul Fattah al-Sisi told the nation that Morsi had to be removed because he had "failed to meet the demands of the people."
A Muslim Brotherhood spokesman said Morsi was under house arrest, together with 12 presidential aides.
Morsi's ouster followed days of street protests in Cairo and other cities.
Mansour's swearing-in took place in a televised ceremony early on July 4, less than 24 hours after army chief Sisi announced plans for the move in a televised speech.
"We were hopeful about achieving a national dialogue that would end the violence and secure stability to the Egyptian people, but the statement by the president last night which was broadcast before the 48-hours deadline was not responsive to the demands of the people and this has forced the armed forces to hold talks with some of the national and political figures as well as the youth," Sisi said. "During the talks, the attendees agreed on a road map which included initial measures aimed at building a strong and united Egyptian society and ending the violence and the conflict in the country."
The military suspended the Islamist-drafted constitution and promised new elections.
Sisi was flanked by political and religious leaders as well as top generals when making the announcement. Pope Tawadros II -- the head of the Coptic Christians -- and Egypt's liberal opposition leader Muhammad ElBaradei made short statements saying the military's "road map" was the best way out of the crisis.
The announcement was met with cheers -- including "Long live the free, independent, Egypt!" -- by the huge crowds gathered in Cairo's Tahrir Square.
'Returning To Square One'
Morsi and his Muslim Brotherhood called the action a "full coup."
Millions of anti-Morsi demonstrators celebrated all night, while troops and armored vehicles deployed in the streets of Cairo and elsewhere, surrounding Islamist rallies.
Clashes, however, erupted in several cities when Islamists opened fire on police, with at least nine people killed.
A video recording purportedly showing Morsi in his office in Cairo was posted late on July 3 to YouTube.
The video, which was removed hours later, appears to have been filmed shortly after Egypt's army overthrew Morsi from power.
In it, Morsi says: "There are deadly attempts to steal this revolution. We are returning to square one, so we can begin all over again and this is something I completely reject and I don't accept this. I am making a direct order to all the forces -- to all the security forces and institutions, the interior ministry, the armed forces -- to have a role in preserving the security of the nation in a clear and concise way. And to avoid bloodshed and to back legitimacy."
The authenticity of the video has not been confirmed.
Meanwhile, the military announced it had shut down three Islamist-run television stations, including one run by Morsi's Muslim Brotherhood.
Morsi became Egypt's first democratically elected president on June 30, 2012, when he won an election following the 2011 Arab Spring revolt that toppled long-time strongman Hosni Mubarak.
Morsi's time in office was marred by continuous political unrest and a collapsing economy.
Mass protests over the weekend that eventually prompted the army's intervention were called in reaction to worsening economic conditions and growing social discontent.
Mansour is a graduate of the Cairo University's law school who also attended a French graduate school for aspiring civil servants. He was appointed to the Supreme Constitutional Court in 1992.
World's Eyes On Cairo
Reaction has poured in from around the globe to Morsi's ouster.
UN Secretary-General Ban Ki-moon appealed for "calm, nonviolence, dialogue, and restraint."
U.S. President Barack Obama said he was "deeply concerned" by the military's move and added that Washington wanted to see a democratically elected government quickly restored there.
Obama said his administration was reviewing foreign aid to Egypt.
The European Union's foreign policy chief, Catherine Ashton, called on "all sides to rapidly return to the democratic process."
Prime Minister David Cameron said Britain was against military intervention, and called for "a genuine democratic transition to take place."
German Foreign Minister Guido Westerwelle said his country was watching the developments in Egypt "closely and with great concern."
He called on all sides to refrain from acts of violence and engage in a democratic dialogue.
"These are the first five minutes of a historic hour for Egypt," Westerwelle said, "and from my heart I hope that Egypt will continue on the road of democracy, and I hope that all the political powers that are squabbling with each other will sit down and talk, that they will find a channel of communication so that this democracy that is so very young does not fail so soon and that it gets the chance it deserves."
Russian Foreign Ministry spokesman Aleksandr Lukashevich said that country was following events and warned Russian tourists to steer clear of volatile areas.
"The development of the situation in Egypt that historically shares friendly relationship of equality and cooperation with Russia is being closely monitored in Moscow," Lukashevich said. "According to the available information, the situation in the resort areas of Egypt remains calm. The tourism infrastructure there continues operating in a standard way. Having said that, however, we consider it necessary to reiterate the recommendation issued earlier by the Russian Foreign Ministry for Russian citizens currently situated in Egypt to avoid the areas of mass protest actions and social tension."
Chinese Foreign Ministry spokeswoman Hua Chunying called on Egyptians to "avoid resorting to violence and resolve the dispute through dialogue and consultation as soon as possible."
Arab nations largely welcomed Morsi's fall. Saudi King Abdullah congratulated Adli Mansour, the head of the country's constitutional court, for being selected Egypt's interim leader.
Gulf Arab states Qatar and the United Arab Emirates also hailed the news of Morsi's ouster.
Syrian President Bashar al-Assad, whose forces are embroiled in a two-year-old civil war with rebels who also include some Islamist radicals, said the upheaval in Egypt was a defeat for political Islam.
Israel has maintained notable silence over events in neighboring Egypt.
Media reports suggest that Prime Minister Benjamin Netanyahu has ordered his cabinet to refrain from commenting on the situation, while closely watching the developments.