The 34-year-old was hit near an intersection at Wyndham Vale in Melbourne's outer southwest on Friday morning and died at the scene.
The Police Association of Victoria said officers called a Melbourne sobering-up centre to assist the man, but they could not attend due to staffing issues.
A spokeswoman for community health organisation cohealth, which runs the Collingwood sobering-up centre, said staff had assessed the man, who was a known client.
She said he was turned away for safety reasons.
"Based on the client's history, it was determined that a referral to the Sobering Centre was not appropriate, and Victoria Police was subsequently informed," she said.
Police said the man had hours earlier been lying on the road at another intersection.
Cohealth's sobering up centre allegedly refused to attend when police found an intoxicated man. (James Ross/AAP PHOTOS)
Two officers spent almost an hour with the man before leaving him at his home in Werribee after being called to another job, police union secretary Wayne Gatt said.
"They tried to get that person to drink, they made that person a bowl of spaghetti and gave that person food," he told reporters on Friday afternoon.
"They rang a sobering-up centre that wouldn't come and then they got called off to a ram raid ... but they did everything they could in the hour or so that they were there."
The union is supporting the officers, whom could not detain or arrest the intoxicated man under "reckless" law changes, Mr Gatt said.
He demanded urgent reform to ensure a safety net to keep the community safe.
"We don't have time to wait for that and indeed the government needs to move quickly to make sure the obvious gap that exists is closed," he said.
Premier Jacinta Allan says Victoria Police and the coroner will investigate the man's death. (James Ross/AAP PHOTOS)
Victoria decriminalised public drunkenness in November, replacing it with a health-based response including outreach teams and a dedicated sobering-up centre operated by cohealth.
The health service is available to publicly intoxicated people who are at heightened risk, but excludes those with serious physical or mental health issues, those who have committed an offence, and pose a safety risk.
Liberal MP James Newbury said the incident was a catastrophic failure of the new system, calling on the premier to explain why the system failed the man.
Premier Jacinta Allan said it was a tragic and terrible incident but would not be drawn on whether the state government held some responsibility for the man's death.
She said the case would be thoroughly investigated by Victoria Police and the coroner.
"I want to understand what has occurred here ... what has led to family grieving today the loss of a loved one," she said.
Police are preparing a report for the state coroner.
The Victorian government committed to decriminalising public drunkenness at the start of an inquest into the 2017 death of Yorta Yorta woman Tanya Day.
Ms Day was arrested for being drunk in a public place and died after hitting her head in a concrete cell at Castlemaine Police Station.
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