Out of the 55 councillors who attended the full council meeting, two independent councillors, 19 Reform councillors and six Conservative councillors voted against the motion.
Fourteen Liberal Democrat councillors, one Conservative, six Green, three Labour and two independent councillors voted in favour.
In addition, two Restore Britain councillors abstained from voting.
Finch said residents wanted “common sense leadership, not a coalition of chaos”.
He said the result of the motion was what the electorate had voted for, and that what had happened in the council chambers would not affect how the authority was run.
He said: “We are in a minority administration and we do have to work with other parties and we’ve shown that from day one.
“To be honest with you, I don’t want to waste any more time on things like that.
“I want to make sure the people get the county council that’s working for them and that’s what I’m committed to do. I don’t want to waste any more time talking about it or going over it, because that’s not what the people voted us in for.”
Chilvers told the BBC that despite the result, the Green Party has not ruled out another challenge to Councillor Finch’s leadership in the future.