Greenaway ruled Winehouse, 27, died from accidental alcohol poisoning and returned a verdict of misadventure at her inquest.
Camden Council told the BBC Greenaway had been appointed "in error" to the post by her husband Andrew Reid, the coroner for inner north London. His actions are being investigated by the Office for Judicial Complaints.
Greenaway resigned this week after it was made public she didn't meet the requirement of practicing as a lawyer in the United Kingdom for five years, even though she had qualified as an attorney in Australia since 1999.
Reid told the British broadcaster he is "confident that all of the inquests handled were done so correctly," but apologized "if this matter causes distress to the families and friends of the deceased."
"The Winehouse family is taking advice on the implications of this and will decide if any further discussion with the authorities is needed," the Winehouse family said in a statement issued to the BBC.