Chantel Moore, 26, was killed Thursday in Edmundston, in the eastern province of New Brunswick after a relative had called police to check for her health.
Police have alleged she was making threats and holding a knife. According to the family, the officer fired five times to subdue her.
Moore's "tragic killing ... during a wellness check has vividly shown all Canadians that Indigenous peoples in Canada continue to face a very different set of circumstances when interacting with the policing and justice systems in Canada", said Congress of Aboriginal People (CPA) National Chief Robert Bertrand.
While officials have opened a probe into Moore's death, the national chief of the Assembly of First Nations (AFN), Perry Bellegarde, said the shooting must be investigated by an impartial third party to determine why lethal force was used and whether race was a factor in the officer's response.
"How does a call for help turn into a call for the coroner? This should never happen," Bellegarde said.
"We need to find out whether race played any role in the police response and whether a less extreme use of force should have been used. This young First Nations mother and daughter did not need to die."
Prime Minister Justin Trudeau spoke Friday about discrimination in Canada before participating in a demonstration in Ottawa against racism and police violence following the death of George Floyd, an unarmed Black man killed by police in the United States.
"Over the past weeks, we've seen a large number of Canadians suddenly awaken to the fact that the discrimination that is a lived reality for far too many of our fellow citizens is something that needs to end," he said.