Several supporters of ousted Egyptian President Muhammad Morsi were said to have been wounded by gunfire as a crowd of several hundred people tried to march toward the military barracks in Cairo where he was being held.
Security forces were cordoning the Republican Guard barracks, but it was not immediately clear who had opened fire.
Opponents of the military intervention to unseat Morsi had called for large rallies after Friday Prayers on July 5.
Marchers turned out in significant numbers in Cairo and other cities.
The military, which ousted Morsi two days earlier, had pledged to guarantee the right to peaceful protest but warned against actions that could threaten national security.
Interim President Adli Mansour has ordered that parliament be dissolved and reportedly named Raafat Shehata to be presidential security adviser, replacing the Morsi appointee who had the post.
Muslim Brotherhood spiritual leader Muhammad Badie appeared at the Cairo rally, saying reports of his arrest were lies. The prosecutor's office had ordered Badie’s arrest on July 4.
The UN's high commissioner for human rights meanwhile warned against "violence...arbitrary detention...[or] illegal acts of retribution" at a "delicate period of transition" for Egypt.
The military toppled Morsi late on July 3 after millions of Egyptians turned out in four days of protests.
Mansour, the head of the Supreme Constitutional Court, was sworn in as interim president one day later.
'Egypt Is Egypt'
There have been concerns of Islamist violence in retaliation for Morsi's removal.
Islam Abdel-Rahman, a member of the Muslim Brotherhood's Freedom and Justice Party's foreign affairs committee, told VOA ahead of the July 5 events that his group would not participate in any military-led political process.
But he also rejected violence.
"We believe in peaceful means of defying this military coup," Abdel-Rahman said, using a word to describe the Morsi ouster that most Western leaders have avoided. "We don't believe in taking up arms or something like this. We still believe this country can be managed by political means."
He added that his country was unlike Pakistan or Algeria, two places where major unrest followed military takeovers.
"Egypt is Egypt and people are very confident and determined that we can give an example of a peaceful challenge to an armed coup," Abdel-Rahman said.
Egypt's army has reportedly closed the Rafah border crossing with the Gaza Strip following attacks on security forces in northern Sinai.
In the most serious of those attacks, one soldier was killed and two were injured, although it was unclear whether the attacks were launched in reaction to Morsi's removal.
The international community has reacted cautiously to the militarily led ouster, with Western governments and major powers focusing on calls for nonviolence.
UN High Commissioner for Human Rights Navi Pillay urged the reestablishment of "the rule of law and a system of government that respects the human rights of all Egyptians -- men and women."
In a statement
, Pillay added that "internationally recognized human rights and freedoms, including freedoms of speech and assembly, need to be upheld during this delicate period of transition."
She also warned against persecution in light of reported detentions of Muslim Brotherhood leaders.
"There should be no more violence, no arbitrary detention, no illegal acts of retribution. Serious steps should also be taken to halt, and investigate, the appalling -- and at times seemingly organized -- sexual violence targeting women protestors," Pillay said.
Based on reporting by AP, Reuters, VOA, RFE/RL, and BBC