The leader of Staffordshire's Conservative group, which ran the council until the May elections, said Cooper should now resign.
Philip White said: "It was inevitable that Ian Cooper would have to go, but Reform have done nothing to address the bigotry that appears to be embedded in their Staffordshire branch.
"Nigel Farage needs to instigate an urgent review of his entire Staffordshire operation. This is the third time since May this has happened with a Staffordshire Reform councillor and that pattern cannot be ignored."
Earlier this week, when allegations about Cooper first surfaced, nine of Labour's Staffordshire MPs called for his resignation.
They said in a statement prior to Friday's announcement: "Ian Cooper has, by sharing this kind of material, exposed his far-right and racist opinions.
"Staffordshire deserves better than this outright bigotry."
Cooper had been formally selected as leader of the authority in May, after Reform UK secured 49 seats on the council.
Councillor Martin Murray has been appointed interim leader of the Reform group at Staffordshire County Council, until the group elects a new permanent leader.
A spokesperson for Staffordshire County Council said proposals to nominate a new leader would need to be agreed at a meeting of full council, and it would work with the administration on the next steps.
The next full council meeting is due on Thursday 11 December.
"Our job will be to continue serving the residents of Staffordshire and delivering services without disruption," the council spokesperson added.
Cooper is not the first of Staffordshire's Reform councillors to face scrutiny over their social media activity.
Last month the county council's cabinet member for highways Peter Mason apologised for his choice of words over posts he wrote about black women and the police.