Former Swindon Borough Council Cabinet member Dale Heenan insists no decision has been made to close the Museum and Art Gallery.
He remains a borough councillor for Covingham and Dorcan.
Cllr Heenan said in a statement on social media: "The amount of time this issue has taken up, and others within my Cabinet Portfolio, has shown me that right now I need to find a better balance between my Council duties, work, family, charitable interests and volunteering for covid vaccination shifts. Following discussions with the Leader of the Council first thing this morning, I resigned from Cabinet."
Cllr Heenan insisted in the statement that no decision about the future of the Museum and Art Gallery, which has been closed to the public since march of last year, had been taken.
In his statement, the councillor repeated his criticism of the Friends of Swindon Museum and Art Gallery, accusing them of making false claims on social media that a decision to make the closure permanent had been made.
Yesterday - 7 July - Cllr Heenan was publicly rebuked by former council deputy leader Russell Holland, who said the relevant information was a matter of public record and added: "What should have been a good natured and important discussion about the future of the Museum has now taken an unfortunate and wholly unnecessary turn."
A change.org petition, Save Swindon Museum and Art Gallery, has so far attracted more than 1,400 signatures.