If it passes, social media platforms could face fines of up to A$50 million, if they fail to verify a user's age.
Musk took to his own social media platform X to slam Albanese government's bill, tweeting that it "seems like a backdoor way to control access to the internet by all Australians."
The world-first legislation, introduced into parliament on Thursday, would also create a legal definition of social media.
It plans to use an age-verification system to enforce a social media age cut-off, some of the toughest social media controls imposed by any country to date.
Musk, who views himself as a champion of free-speech, has previously called out Australia's Labor government over its social media policies and had called it "fascists" over its misinformation law.
In April, X went to an Australian court to challenge a cyber regulator's order for the removal of posts about the stabbing of a bishop in Sydney, leading to Albanese labelling Musk an "arrogant billionaire".