The so-called “conscience rule” was due to come into effect on 22 July, but has been postponed because of a legal challenge in California.
Opponents warn it would endanger patients, particularly women and LGBT+ people seeking access to contraception, HIV treatment or abortion.
The final court ruling on the issue will now take place on 22 November.
What does the rule say?
The rule, known in full as Protecting Statutory Conscience Rights in Health Care: Delegations of Authority, was announced by the US Department of Health and Human Services (HHS) in May and is a priority for religious conservatives.
Under the measure, institutions receiving government funding would be required to certify that they comply with existing federal laws protecting medical workers who object to some procedures.
If an institution does not allow its workers to exempt themselves, its funding would be withheld.
Who is challenging it?
San Francisco City Attorney Dennis Herrera filed a lawsuit against the rule within hours of it being announced, arguing that the HHS had overstepped its bounds.
On Saturday the two sides announced that they had mutually agreed to the delay, and that it had been granted by the judge overseeing the case.
In a statement, Mr Herrera said: “The Trump administration is trying to systematically limit access to critical medical care for women, the LGBTQ community, and other vulnerable patients. We’re not going to let that happen.”
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He added: “Hospitals are no place to put personal beliefs above patient care. Refusing treatment to vulnerable patients should not leave anyone with a clear conscience.”
San Francisco would lose about $1b (£800m) in federal funding under the rule, the statement noted.
Similar lawsuits challenging the rule have been filed in other states and cities, and by California Attorney General Xavier Becerra.