Marama Davidson has become the sole leader until there's another vote.
Shaw could put his name forward again as a contender for the role when the party votes to fill the position within five weeks, but yesterday said he would need to consider that.
He confirmed that Prime Minister Jacinda Ardern has kept him on as Climate Change Minister.
Bradford said today that Shaw's weak leadership had divided the Greens, which must compromise with the Government.
That could hurt the party's chances at the next election, she said.
She said there was talk of Chloe Swarbrick's suitability for the co-leader role, but it's not clear whether she's considering a leadership challenge. Swarbrick did not answer RNZ's calls today.
Nine MPs contacted by RNZ declined to comment if they would be going for the position.
Strategy 'not necessarily supported by everybody'
Former Greens MP Catherine Delahunty said Shaw may have been voted out of the co-leader position for not taking a strong enough stance on climate change.
She says dissatisfaction with Shaw may stem from his focus on consensus within Parliament in his role as Climate Change Minister.
Instead, the Greens should be the strongest and most vocal party on climate change, and call the Labour party to account, she believed.
"What he has believed to be the best strategy is not necessarily supported by everybody because it's not resulting in pressuring Labour to take stronger action - in fact, it's seen as very weak by many of us involved in activism, and I think obviously by some of the party members as well."
Delahunty said it was important to focus on the vision other potential leaders offered, rather than speculating on which individual might replace Shaw.
She said the Greens needed transformative policies to stop climate change, defend the vulnerable in society, and improve social justice.
"I can't encapsulate that and say there's one individual right now that I would support," she said.
"It's more about if James doesn't stand again, it opens up an opportunity for somebody to come out and show some vision, and we really need to hear that. It's got to be different from a middle-class, middle-age party just propping up the Labour government."
Ejection 'undemocratic'
A Green party member said it was undemocratic and bad that Shaw had been ejected from his position.
A party member told RNZ the party's poorly-thought through rules allowed a minority to destabilise Shaw's leadership, potentially to the point it becomes untenable.
Shaw was voted out after 32 of 107 delegates voted to vacate his position - more than the 25 percent threshold necessary under the Greens' rules.
This member said it was not democracy or consensus, but giving veto power to a vocal minority.
They said the result was not the wishes of the vast majority of the party members.
Not easy being Green: Shaw to seek party feedback
Shaw says he wants to sound out the party's membership before confirming whether he would run for the position of co-leader again or not.
He indicated he was inclined to put his hand up again, but needed to find out if that was what members wanted first.
"The question that I've got is if the nominations close and I'm the sole nominee again, will the vote go exactly the same way or will it change? And so that's really my one hesitation," he said.
Green Party co-leader Marama Davidson, whose position was reconfirmed by the delegates votes, wouldn't say concretely whether she wanted Shaw to be her co-leader again.
However, she said he had worked hard in the role.
Davidson said she was shocked and saddened, but wouldn't get ahead of the party's nominations by saying whether she'd back Shaw.
"I simply cannot pre-empt that conversation but what I can do without damaging the process of the Green Party is be really clear about how he's slogged his guts out behind the scenes."
Nominations for Shaw's spot would be open for a week, with any election to be held in a month.
