Anthony Albanese said Australia was the first nation to write to UK Prime Minister Keir Starmer to support the push to remove the disgraced former royal from the line of succession to the British and Australian crowns.
The ex-prince was arrested over the alleged leaking of secret documents to dead pedophile Jeffrey Epstein. He was released as investigations continue.
All 15 countries in the Commonwealth realm, including New Zealand, Canada, and Papua New Guinea, have to agree to remove Mountbatten-Windsor from the line of succession.
"These are very serious allegations," Mr Albanese told ABC radio on Tuesday.
"Australia likes being first and we have made sure that everyone knows what our position is, and we'll be writing today to the other realm countries informing them of our position."
Mr Albanese said the move was important even before the investigation is complete.
"Already, Andrew Mountbatten-Windsor, the fact that we're calling him that rather than Prince Andrew indicates that there's already been moves along those lines," he said.
He wrote to Mr Starmer: "I agree with His Majesty that the law must now take its full course and there must be a full, fair and proper investigation.
"These are grave allegations and Australians take them seriously."
The prime minister said he is not planning a referendum for Australia to leave the monarchy and become a republic.
"I haven't changed my position, I'm a republican," he said.
Mountbatten-Windsor is eighth in line to the throne, which means he could technically become Australia's head of state.
But such a scenario is unlikely because it would require the death of King Charles, Prince William, Prince Harry and five royal children.
The UK is considering removing Mountbatten-Windsor from its line of succession as a symbolic move after his arrest.
Mountbatten-Windsor was arrested on his 66th birthday after a trove of emails released in the US suggested he leaked confidential documents to Epstein while serving as the UK's trade envoy.
The former prince has also been dogged by accusations he sexually abused Virginia Roberts Giuffre and other women when they were teenagers - claims he denies.
