The Vatican announced the resignation in a one-line statement. It was the latest in a string of cases of alleged misconduct against bishops to come to light this year, following the scandal of ex-Cardinal Theodore McCarrick that exposed how bishops have largely avoided sanction for improper behaviour.
The current archbishop of Los Angeles, Most Rev Jose Gomez, said the archdiocese was made aware of the claim in 2005.
Rev Gomez said prosecutors declined to bring charges, but that the archdiocese forwarded the complaint to the Vatican office that handles sex abuse cases.
Rev Gomez said that office, the Congregation for the Doctrine of the Faith, imposed precautionary measures against Salazar and that a further investigation by the archdiocese's independent review board found the allegation to be credible.
Rev Gomez said Salazar, 69, has "consistently denied any wrongdoing". The archdiocese said it had received no other allegations against Salazar.
"These decisions have been made out of deep concern for the healing and reconciliation of abuse victims and for the good of the church's mission," Rev Gomez told the Los Angeles faithful in a letter.
"Let us continue to stay close to the victim survivors of abuse, through our prayer and our actions."
Rev Gomez said that the alleged misconduct occurred while Salazar was a parish priest in the 1990s, and that the claim was never directly brought to the archdiocese. Other details were not released.
The resignation comes during a year in which the clerical abuse scandal has exploded anew in the US and elsewhere.
In July, Pope Francis removed McCarrick, the retired archbishop of Washington, as a cardinal after a US church investigation determined that an allegation he groped a minor in the 1970s was credible.
Subsequently several adult seminarians have said he pressured them to sleep with him.
McCarrick has denied the initial allegation, and through a lawyer has said he looks forward to invoking his right to due process.
The McCarrick scandal exposed the loopholes in how the church treats allegations of misconduct against bishops, who are answerable only to the pope.
Bishops have largely escaped the same scrutiny as ordinary priests accused of misconduct in the decades-long sex abuse scandal, and until recently they have rarely been sanctioned or removed for covering up for abusers.
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