Bryce made the comments at a speech in Sydney yesterday evening in which she outlined a future vision for Australia in which she imagined a nation "where an ethic of care guides the way we lead".
"Where the young, the elderly, the indigenous, the newly arrived, people with disabilities, are treated with dignity and respect," she said.
"Where every child is given the opportunity of a good education from their very early years; where women's contributions to civil society, the workplace, the economy, the family and home are valued equally with men's... Where people are free to love and marry whom they choose;
Bryce, the nation's first woman governor-general since she was appointed by then prime minister Kevin Rudd in 2008, was praised by republicans and advocates for same-sex marriage for her speech.
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But her comments, reportedly the first by a sitting governor-general in favour of a republic, were criticised by David Flint, national convenor of Australians for Constitutional Monarchy as "a great pity".
"The constitutional system requires that the Crown be above politics so that when the representative of the Crown... speaks, they shouldn't be talking about politics," he told the Australian Broadcasting Corporation.
"So we've got commentators everywhere on these issues, couldn't she have left them alone until she was out of the office?"
Chair of the Australian Republican Movement, Geoff Gallop, said while Bryce had served with distinction as the queen's representative, it would send a powerful message about the country if its head of state was drawn from the Australian people.
Prime Minister Tony Abbott, a staunch monarchist, said Bryce was entitled to express her personal views, adding that she did so with grace and style.