The Lord Chancellor and the Lord Chief Justice had indicated their intention to remove Dr Andrew Reid from judicial office following the investigation.
But despite being entitled to challenge their decision, the senior coroner decided to stand down.
During her time as deputy assistant coroner, Ms Greenaway conducted 12 inquests in Camden, but worked largely from Poplar Coroner's Court in east London.
According to British law, Ms Greenaway, who also worked as a nurse in Australia, would have needed either five years with the Law Society or five years of experience as a qualified medical practitioner.
She met neither set criteria.
More top news
The one-year-old Yorkshire Terrier was attacked and killed while being taken for a walk.
The lifeboat crew carried the pair into their boat and dropped them off back at Tower Bridge as people watched from the riverside.
The huge painting took two graffiti artists thousands of hours to complete in east London.