From 31 August, NHS and social care staff will also not be offered lateral flow tests unless they fall sick.
Free testing for the general public ended in April in England, but continued in some high-risk settings.
The government said UK Covid rates have fallen, meaning most testing could be paused.
It said individuals would continue to be protected through vaccination and antiviral treatments and it would keep the situation under close review.
The Department of Health said that, under the new rules, routine asymptomatic Covid testing of staff and existing patients or residents would be paused in:
* the NHS
* adult social care and hospice services
* parts of the prison estate and some places of detention
* certain domestic abuse refuges and homelessness settings
All new admissions into care homes and hospices will continue to be tested for Covid, as will patients with weakened immune systems who are admitted to hospital or transferred between wards.
Staff and patients will also still be tested if they have Covid symptoms such as a fever, persistent cough or loss of smell.
Health Secretary Steve Barclay said: "This reflects the fact case rates have fallen and the risk of transmission has reduced, though we will continue to closely monitor the situation and work with sectors to resume testing should it be needed."
He said the forthcoming autumn booster vaccine programme, along with antiviral treatments, would protect those at greatest risk from severe Covid.
Covid infection rates have been falling across England over the last month and the government said it expects prevalence to remain low following the most recent wave, caused by the BA.4 and BA.5 variants of Omicron.
Dr Susan Hopkins, the UK Health Security Agency's chief medical adviser, said: "Covid case rates and hospitalisations are on the decline, demonstrating the positive impact of the vaccines, which remain our best form of defence. The data from our surveillance shows prevalence is low and decreasing, and we will continue to monitor this data closely."
The Royal College of Nursing is concerned about the pause.
Patricia Marquis, RCN Director for England, said: "Cases of COVID-19 may well be falling but this virus has still not gone away, and it is vital that there is continued vigilance to ensure patients and nursing staff are not put at risk.
"Nursing staff must continue to have access to free testing and high-quality personal protective equipment.
"We have all come a long way and must not risk any backwards step when health services are already under enormous pressure."
The advice in Scotland,Wales and Northern Ireland differs, but currently recommends NHS staff take regular lateral flow tests even if they do not have symptoms.