Scott Morrison is under increasing pressure to publicly rebuke a government backbencher for spreading misinformation about coronavirus.
Liberal MP Craig Kelly has been promoting unproven cures on social media and last week likened mask mandates for school students to child abuse.
With Mr Morrison back at work on Monday after a week-long holiday, Labor is demanding he censure his partyroom colleague.
"Labor condemns Craig Kelly for spreading misinformation in a pandemic. It's on all of us as leaders to call it out," Opposition Leader Anthony Albanese said.
"Now it's your turn, PM."
Three in four people surveyed by the Australia Institute believe Mr Morrison should publicly criticise his MP.
The left-leaning institute's deputy director, Ebony Bennett, says the prime minister has a responsibility to set the record straight.
"By remaining silent on misinformation spread by members of his own government the prime minister is putting the health of Australians and the health of our democracy at risk," Ms Bennett said on Monday.
"Vaccinating the Australian population against COVID-19 will be one of the largest peacetime operations in Australian history.
"If MPs who spread misinformation like Craig Kelly have the tacit endorsement of the prime minister it will only jeopardise and undermine the success of the public health effort."
More than half the 1003 people surveyed also want the prime minister to condemn Donald Trump for his role in inciting the US Capitol riots.
"The prime minister's lack of condemnation of Trump's role in the deadly insurrection is a failure to defend the most basic principles of democracy," Ms Bennett said.
"Prime Minister Scott Morrison should join other world leaders including Boris Johnson, Angela Merkel and Justin Trudeau who have criticised or condemned United States President Donald Trump's role in inciting the insurrection."
Senior cabinet minister Simon Birmingham leapt to the prime minister's defence.
"Scott Morrison was out there pretty quickly and condemned the violence pretty quickly," he told ABC radio.
"I well and truly condemn not only the violence or the incitement of it but also what we have seen in the US in particularly recent months, an erosion of confidence in democratic systems, principles and values. It's horrific."
Senator Birmingham denied Mr Morrison had "pulled his punches" with Mr Trump.
"We are always cautious about running commentary as the Australian government on the political leaders of foreign governments," he said.
"And Donald Trump, whether people like it or not, is still the US president for a few more days, and then we very much look forward to working with the Biden administration and then president Biden once he takes over.
"So, it's in Australia's interest for us always to be mindful that little has advanced for Australia by it, just running endless commentary on the rest of the world."
During a stint as acting prime minister last week, Michael McCormack declined to rebuke Mr Kelly and refused to condemn Mr Trump.
Australian Associated Press