Jacob Blake was undergoing surgery to try to repair nerve damage, his cousin said.
"He's paralyzed from the waist down... He hasn't woken up yet but they're saying it's a 50/50 chance he'll walk away," Herman Poster told the Daily Beast news site.
It was the most recent in a series of allegedly unjustified police shootings and killings of African Americans that have sparked outrage in the United States.
Blake was shot on Sunday by a white policeman while getting into a car that held his three children after trying to break up a domestic dispute, according to civil rights attorney Benjamin Crump, who is representing the Blake family.
Jacob Blake's father, also named Jacob, told the Chicago Sun Times "What justified all those shots?... What justified doing that in front of my grandsons? What are we doing?"
A bystander video shows police shooting seven times at Blake, 29, at close range as he tried to enter the car.
Officials say police had been called to a domestic disturbance, but have not said why the two officers had their guns pulled.
Late on Monday police wearing military-type gear fired tear gas at hundreds of protesters who snubbed a citywide curfew to demonstrate over Blake's shooting.
The protesters, chanting "No justice, no peace" and "Say his name - Jacob Blake," shot fireworks and lobbed water bottles at the police.
Sherese Lott, 37, said she attended with her two daughters and a cousin to protest "because we want change".
She questioned why the officers did not tackle or Taser him if they wanted to arrest Blake, and said police need to be held accountable.
"If I killed someone I'd be convicted and regarded as a murderer. I think it should be the same for the police."
A couple, who only gave their names as Michelle and Kalvin, had brought their seven-year-old daughter and eight-year-old son with them.
"I want my kids to see how change happens and am here so nothing like this ever happens to them," Michelle said.
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Police under scrutiny
The footage and the lack of immediate explanation provoked painful memories of the police killing of African American George Floyd on May 25 in Minneapolis, which led to protests spreading in the US and abroad.
As the two officers involved were placed on administrative leave, Wisconsin authorities launched a probe into the shooting.
Democratic presidential candidate called on Monday for a "full and transparent investigation".
Wisconsin's Lieutenant Governor Mandela Barnes said the shooting of Blake reflected a pattern of police violence nationwide against African Americans.
Barnes said Blake "was actually trying to de-escalate a situation in his community but the responding officer didn't feel the need to do the same".
Crump, who also represents the families of George Floyd and other black victims of police mistreatment, said Blake's family had been able to visit him in the hospital.
"They are relieved that he is in stable condition," Crump told ABC's Good Morning America show.
"However he is still in intensive care and he is still facing more surgeries in the future," he said.
Crump said it was thought that Blake was hit by four of the seven shots fired by one of the two officers.
He said he had no indication or witness accounts that Blake was armed or had an altercation with the police.
"He was walking away. He wasn't posing a threat to them," Crump said.
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