"We will not stand by silently watching the destruction of the country and the people or the torching of the nation and terrorising of the citizens," he said.
As he spoke, the Cabinet he put in office was discussing whether to revive the formal ban on the Brotherhood which was lifted after the fall of the former military leader, President Hosni Mubarak.
The Interior Ministry said the 36 prisoners had taken an officer hostage and died after suffocating on tear gas.
A legal source told Reuters the men died from asphyxiation in the back of a crammed police van.
A further round-up was already under way of Brotherhood officials across the country, including more than 100 in the southern province of Assiut alone.
Those Brotherhood leaders not already under arrest or on the run said the army was pushing Egypt into a "Syria situation" in which even if they urged peaceful resistance, protesters and radical Islamists outside their control might take up arms.
The civil war in Syria began with the army and police firing on largely peaceful demonstrators.
Armed resistance on a large scale began after the number of dead hit four figures.
"It's as if there is someone who wants to push the country into this Syrian scenario," said Saad Emara, who was deputy chairman of the National Security Council of the Upper House of Parliament under Morsi.
"The number of bodies from each massacre is far bigger than the massacres there were in Syria," Emara said.
There were signs that Egypt's traditional Western backers were finally starting to lose patience with the army's tough line, or at least the rising number of dead. A further 79 people died in violence on Sunday, the authorities said, on top of more than 800 as protests were broken up on Thursday and Saturday.
The Brotherhood says that more than 2000 people have been killed.
The European Union, in its most strongly worded statement yet, said the killings "could not be condoned".
Calls for democracy and fundamental rights "cannot be disregarded, much less washed away in blood", said a joint statement by Jose Manual Barroso and Herman van Rompuy, presidents of the European Commission and Council.
Two leading Republican senators from the United States, John McCain and Lindsey Graham, whose peace mission to Cairo this month was directly snubbed by Sisi, called for US aid to be suspended.
The US backed Egypt's military rulers for decades, and continues to face pressure from key allies such as Israel and Saudi Arabia to do so.
Again a lone figure faces down the tanks, but this time the bullets fly
Shot at close range as he stood, arms aloft, in front of the tanks, Ahmed Hashem Rashwan had the makings of a martyr to Egypt's military coup.
References to Tiananmen Square have proliferated in today's Egypt, and as a video camera rolled unseen behind him, he appeared to be recreating a celebrated image from 24 years ago.
The result was startlingly different. Shots are heard ringing out; the slender figure in a white shirt crumples and folds over, rising from the ground and then collapsing again, writhing before becoming still.
But tracked down yesterday, he was, remarkably enough, not only still alive but after two rounds of surgery, even able to speak about his actions.
"I was hit by two bullets, one in my stomach and one in my leg," he said. "No one did anything to provoke them, but it seems that they are intent on forcing silence on everyone, that anybody who says no to the coup will be killed or arrested. They are accusing us of being terrorists."
Rashwan, 34, an electronics engineer, was part of a crowd protesting in Ismailia, a town on the Suez Canal southeast of Cairo.
The journalist who shot the video, Abdullah Shocha, who works for a pro-Islamist television station, caught the image of Rashwan, whom he did not know before, approaching the tanks with his arms above his head in a gesture of surrender.
Rashwan, who is not a member of any political or Islamist group, said he had reacted to the first sounds of shooting. "I approached them to send a message that we were peaceful but they shot at me," he said.
"I'm not a complete supporter of the Muslim Brotherhood, but I'm afraid that the country is returning back to military rule and that is not a good future for my son, who I want to live a better life."
