The movement's chair Peter FitzSimons said the country should no longer delay talks about moving away from the monarchy.
"Rule by birthright, a literally born-to-rule English sovereign, has no place in a democratic, egalitarian Australia," he said on Friday.
"The notion is as foreign to Australian values as the monarchy itself. Nor should anyone be forced to pledge allegiance to a foreign king or head of state."
Federal parliamentarians allocated Friday for both chambers to offer condolences to the late Queen and pay tribute to King Charles III.
But some used their speeches to weigh in on the republic debate.
Greens leader Adam Bandt said the death of the Queen and the accession of King Charles III should lead to talks on changing the system of government.
"Now we have a king. We did not elect this man. Nor did we as a people truly consent to be governed by him. We have, respectfully, unfinished business," he told parliament.
"The head of state of this country should be chosen by the people, for the people and from the people.
"We should respect the civility with which Elizabeth Windsor oversaw the drawdown of what was once the British Empire and take the cue to grow up and move out."
Mr Bandt said people could offer condolences about the monarch and have conversations about whether a constitutional monarchy was right for Australia.
Liberal senator Gerard Rennick said a republic push would not benefit Australia, with more pressing issues facing the nation.
"If there is to be a debate about our government in this country, it needs to focus on the dysfunctional relationship between our state and federal governments … Australia does not need to entertain a symbolic debate about a ceremonial head of state," he said.
"While our health system and many other essential services are falling apart, there are much more important issues that governments should be dealing with."
Greens senator Sarah Hanson-Young expressed her condolences to the royal family, but said now was the time to move forward with an Australian head of state.
"King Charles III is not our choice. The Australian people didn't get to choose and we should have," she told the Senate.
"Our head of state should be one of us, an Australian."
The assistant minister for the republic Matt Thistlethwaite did not mention the issue in his condolence speech.
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