Aston Villa had announced on Friday that they intended to pursue legal proceedings against the former Liverpool defender over what they described as "unacceptable insults" made during an appearance on the Stick to Football podcast.
The dispute centred on Carragher's comments about Unai Emery and the possibility of the Villa manager replacing Michael Carrick at Manchester United.
Carragher suggested Emery would be attracted by the opportunity to manage another major European club, arguing that football operates according to a hierarchy in which bigger teams can take the best players and managers from those below them.
"There is a chain in football, whether you like it or not," he said.
"So Real Madrid will take Manchester United's best player, they did it with [Cristiano] Ronaldo. And there will be teams below Aston Villa, and they'll take their players or manager.
"Aston Villa will never be Manchester United."
The comments prompted an angry reaction from Villa supporters, with Carragher subsequently telling those who criticised him to "shut up".
The club's response was unusually strong, with owner Nassef Sawiris announcing that Villa would initiate legal action over the remarks.
However, Carragher issued an apology on Saturday, acknowledging that his choice of words had been inappropriate and that his intended humour had not been understood in the way he expected.
"I'd like to apologise to Aston Villa, the owners and their fans," he wrote on social media.
"My language this week was clumsy and wrong.
"I can assure Villa fans it was tongue in cheek but I can understand it's not come across that way, so apologies."
Villa have now decided not to proceed with the legal threat following the apology. The club had not formally initiated proceedings before withdrawing its intention to do so.