"Our thoughts are with her family and all who loved her. Now Australia must move forward," said Bandt on Twitter.
"We need Treaty with First Nations people, and we need to become a Republic," he added.
Bandt was however accused by some supporters of being disrespectful by bringing up the issue within hours of the monarch’s demise.
"Absolutely disgusting. You can't even wait at least 6 hours before pushing a policy agenda. Disgraceful," one user tweeted in response.
Australia is a constitutional monarchy, with the British monarch as head of state, although the role is largely ceremonial. But a debate has been going on for some time on whether the country still needs to keep a distant monarch as the head of state. A 1999 referendum in Australia on becoming a republic was narrowly defeated.
When asked about the issue, Australian Prime Minister Anthony Albanese said: "Today's a day for one issue and one issue only, which is to pay tribute to Queen Elizabeth II."