A strongly-worded letter sent to children's minister Nadhim Zahawi and Justice minister Edward Argar, said the institution was "irredeemable".
Signatories include former children's commissioner for England Maggie Atkinson; and charity bosses Deborah Coles, executive director of Inquest; John Simmonds, director of policy, research and development at charity CoramBAAF; and Kathy Evans, chief executive, of Children England.
Staff at Medway STC were found to be continuing to use "force and restraint" to deal with "children's passive non-compliance", according to a joint inspection report published last month.
This was despite such practices being exposed in a BBC Panorama investigation three years ago, which led to charges being brought against eight members of staff.
Since then, the STC has received two inadequate inspection judgments in June 2016 and March 2017 from the Care Quality Commission, HM Inspectorate of Prisons, and Ofsted.
A serious case review published by Medway Safeguarding Children Board in January this year laid bare failings by agencies surrounding the centre, including the local authority, Kent Police, Medway NHS Foundation Trust and Barnardo's.
It concluded that use of restraint techniques involving inflicting pain "appeared to enable an environment of increased risk of abuse to children".
The letter to ministers, dated yesterday, continues: "Three years after a BBC Panorama programme showed children being subjected to physical and emotional abuse, an Ofsted inspection report last week stated that pain-inducing restraint is still being used in the centre for non-compliance.
"This is a breach of children's right to protection from inhuman and degrading treatment or punishment, under Article 3 of the European Convention on Human Rights.
"Other ministers before you have failed to take robust action to protect vulnerable children in custody.
"We urge you to close down this institution and move children to settings where they will receive safe and skilled care."
In October 2012, the government announced that Medway STC would be converted into a secure school for young offenders in 2020, with an aim to place greater focus on the education and rehabilitation of young offenders.
Article 39 director Carolyne Willow, one of the letter's signatories, said changing the name of the institution and giving it "a small refurbishment" would not address its systemic failings.
"If this child prison was a children's home or a school, its gates would have been closed shortly after the shocking BBC Panorama exposé of physical and emotional abuse three years ago," said Willow.
"Whenever there is institutional failure stretching back many years, there comes a point when someone in charge has to take decisive action and say ‘enough is enough'."
"The SCR and the latest inspection report show deep fault lines in protecting children.
"Ministers must stand up to the false optimism that the institution can reform itself, and local and national agencies will properly scrutinise and protect children."
The demand is also signed by Just for Kids Law chief executive Enver Solomon, Liberty acting director Corey Stoughton, and National Association for Youth Justice vice-president Barry Anderson.
A Ministry of Justice spokesperson said "significant improvements" had been made at Medway STC since it took over the running of the site from G4S in July 2016, following the allegations highlighted by Panorama.
"The safety of all young people in custody is our absolute priority," she said.
"The latest Ofsted report recognises that standards at Medway continue to improve, including oversight of the use of force, safeguarding and the quality of teaching and education - though we are clear that there is still more to do.
"We are fundamentally reforming youth custody, increasing frontline staff in public sector Young Offender Institutions by 20 per cent, improving training for officers working with young people, and we recently announced a £5m investment, which will see a new secure school established at Medway."
The most recent inspection report of the STC found it "requires improvement to be good".
The Department for Education has also been contacted for a response.