Emery, who joined Villa in 2022, has transformed the club's fortunes.
Posting on social media on Saturday, Carragher retracted his previous remarks and described his choice of words as clumsy and wrong.
Villa never formally initiated legal action against Carragher before deciding to end the threat entirely.
Aston Villa have dropped their threat of legal action against Sky Sports pundit Jamie Carragher.
Carragher apologised on Saturday for using"clumsy and wrong"language, and said his comments were"tongue in cheek".
Though Villa never initiated legal action against Carragher, they have now ended their threat to do so following his apology.
"There is a chain in football, whether you like it or not,"said Carragher.
Posting on social media on Saturday, Carragher retracted his comments.