Ms Turei was defiant this afternoon after a lengthy caucus meeting.
She said there was nothing that would make her step down as co-leader before the September 23 election.
Mr Gillies said he was "fully supportive" of Ms Turei remaining as co-leader.
"Her frank admission of welfare fraud as a young, struggling solo mother was courageous and, along with the Greens' families policy that she launched, has sparked an intense national debate on what it means to live in poverty, and what role an affluent society such as ours should play in order to end it.
"Personal judgments about things someone did in their formative youth are irrelevant, and the electorate at large will form its own opinion."
Mr Gillies said there was "no doubt" the Greens' social policies had received a lot of media air time in recent weeks, and there was plenty more of the election campaign to run.
"The Greens are absolutely focused on clean water, preserving our environment and creating a country where all families can thrive. No other party in Parliament is as committed to reducing our national carbon emissions, and protecting future generations from the impacts of climate change."