The military and security forces returned the fire. Military helicopters flew over the area.
Egypt indefinitely closed its nearby border crossing into the Gaza Strip after the assault, sending 200 Palestinians back into Gaza, said Gen. Sami Metwali, director of Rafah passage.
Extremist Islamic militants have gained strength in the Sinai over the past two years since a security breakdown that accompanied the 2011 uprising that forced Morsi's predecessor, Hosni Mubarak, from power. Egypt's military and police have been battling to contain them.
The military forced Morsi out Wednesday after millions of Egyptians turned out in four days of protests. After its top leaders were targeted with arrest warrants, the Muslim Brotherhood hotly rejected an appeal by the military to take part in forming a new regime.
A military statement late Thursday appeared to signal a wider wave of arrests was not in the offing. A spokesman, Col. Ahmed Mohammed Ali, said in a Facebook posting that that the army and security forces will not take "any exceptional or arbitrary measures" against any political group.
The military has a "strong will to ensure national reconciliation, constructive justice and tolerance," he wrote. He spoke against "gloating" and vengeance, saying only peaceful protests will be tolerated and urging Egyptians not to attack Brotherhood offices to avert an "endless cycle of revenge."
The Brotherhood charged the military staged a coup against democracy and said it would not work with the new leadership. It and harder-line Islamist allies called for a wave of protests Friday, naming it the "Friday of Rage," vowing to escalate if the military does not back down.
Brotherhood officials urged their followers to keep their protests peaceful. Thousands of Morsi supporters remained massed in front of a Cairo mosque where they have camped for weeks, with line of military armored vehicles across the road keeping watch.
"We declare our complete rejection of the military coup staged against the elected president and the will of the nation," the Brotherhood said in a statement, read by senior cleric Abdel-Rahman el-Barr to the crowd outside the Rabia al-Adawiya Mosque in Cairo.
"We refuse to participate in any activities with the usurping authorities," the statement said, while urging Morsi supporters to remain peaceful. The Rabia al-Adawiya protesters planned to march Friday to the Ministry of Defense.
The Brotherhood denounced the military's shutdown of its television channel, Misr25, its newspaper and three pro-Morsi Islamist TV stations. The military, it said, is returning Egypt to the practices of "the dark, repressive, dictatorial and corrupt ages."
Morsi has been under detention in an unknown location since Wednesday night, and at least a dozen of his top aides and advisers have been under what is described as "house arrest," though their locations are also unknown.
Besides the Brotherhood's top leader, General Guide Mohammed Badie, security officials have also arrested his predecessor, Mahdi Akef, and one of his two deputies, Rashad Bayoumi, as well as Saad el-Katatni, head of the Brotherhood's Freedom and Justice Party, and ultraconservative Salafi figure Hazem Abu Ismail, who has a considerable street following.
Authorities have also issued a wanted list for more than 200 Brotherhood members and leaders of other Islamist groups. Among them is Khairat el-Shater, another deputy of the general guide who is widely considered the most powerful figure in the Brotherhood.
The arrest of Badie was a dramatic step, since even Mubarak and his predecessors had been reluctant to move against the group's top leader. The ranks of Brotherhood members across the country swear a strict oath of unquestioning allegiance to the general guide, vowing to "hear and obey." It has been decades since a Brotherhood general guide was put in a prison.
Badie and el-Shater were widely believed by the opposition to be the real power in Egypt during Morsi's term.
The National Salvation Front, the top opposition political group during Morsi's presidency and a key member of the coalition that worked with the military in his removal, criticized the moves, saying, "We totally reject excluding any party, particularly political Islamic groups."
The Front has proposed one of its top leaders, Mohammed ElBaradei, to become prime minister of the interim Cabinet, a post that will hold strong powers since Mansour's presidency post is considered symbolic. ElBaradei, a Nobel Peace laureate who once headed the U.N. nuclear watchdog agency, is considered Egypt's top reform advocate.
"Reconciliation is the name of the game, including the Muslim Brotherhood. We need to be inclusive," Munir Fakhry Abdel-Nour, a leading member of the group, told The Associated Press. "The detentions are a mistake."