Adly Mansour also said he had ordered an investigation into the deaths.
The Muslim Brotherhood says its members were fired on as they staged a sit-in for ousted President Mohammad Morsi, while the army said "terrorists" tried to storm the barracks.
The Brotherhood's political wing meanwhile called for an "uprising".
The Freedom and Justice Party (FJP) - which took nearly half the seats in historic parliamentary elections held in late 2011 and early 2012 - urged Egyptians to revolt against "those trying to steal their revolution with tanks".
It also urged the international community to intervene to "stop further massacres" and prevent Egypt becoming "a new Syria". The movement has accused the army of staging a coup.
Mr Morsi, an Islamist and Egypt's first freely elected leader, was ousted by the army last week after mass protests.
The hardline Salafist Nour party - which had supported Mr Morsi's removal - said it was withdrawing from talks to choose an interim prime minister, describing the shooting incident as a "massacre".
The killings follow an incident in the same location on Friday in which three people died and dozens were wounded as troops fired on crowds. Mr Morsi is believed to be held at the barracks.
In a separate development, the grand sheikh of al-Azhar University, Ahmed al-Tayeb - seen as the highest authority in Sunni Islam - warned of civil war and said he was going into seclusion until the violence was over. (+ videos) » | Monday, July 08, 2013
