A Bill Leak cartoon published in The Australian newspaper on Thursday depicts an Aboriginal man who has forgotten his son's name.
Indigenous groups said the cartoon was "ugly, insulting and embarrassing".
But the paper's editor said the cartoon brought a "crucial issue" into the public domain.
In the cartoon, a police officer is shown bringing an Indigenous child to his father, saying: "You'll have to sit down and talk to your son about personal responsibility."
The father, who is barefoot and holding a beer can, asks: "What's his name then?"
'A fact of life'
The cartoon comes in the wake of debate about the Northern Territory's juvenile justice system and high incarceration rates among Indigenous youth.
It appears to be a response to comments from Indigenous leader Noel Pearson, who said this week that Aboriginal people needed to take more responsibility for the behaviour of their children.
The SNAICC, a non-governmental group for Indigenous children and families, called the cartoon "disgusting, disrespectful, and hurtful", adding: "Those involved in publishing such a clearly racist cartoon should be ashamed and should issue a public apology to all Australians."
The New South Wales Aboriginal Land Council criticised the cartoon, saying it was "embarrassing for Australia's national newspaper to publish it".
"Sadly racism and discrimination is a fact of life for Aboriginal people who have lived on and cared for this country for more than 60,000 years," the statement said.
"It is time the decision-makers at The Australian accept personal responsibility for the hurt they have caused Aboriginal people today."
The Australian newspaper typically takes a right-wing position on social affairs, favouring individual responsibility and free-market economics over government spending and intervention.
But it dedicates substantial resources to Indigenous affairs and has in the past won praise from Aboriginal leaders for its coverage.
The newspaper's editor-in-chief, Paul Whittaker, stood by the cartoon, saying too many people skirted around issues in Indigenous affairs.
"Bill Leak's confronting and insightful cartoons force people to examine the core issues in a way that sometimes reporting and analysis can fail to do," he said in a statement.