Sterling confirmed he had turned down a new deal at Anfield worth £100,000 a week and described links with Arsenal as "quite flattering" in a BBC interview aired on Wednesday.
That prompted an angry response from Carragher, who labelled Sterling's decision to go public with his feelings as "a PR disaster" and criticised the 20-year-old for a "woefully misjudged interview".
Addressing Sterling's assertion that he was attempting to "kill off the stories and change how I am perceived," Carragher - writing for the Daily Mail - said: "They [Sterling and his advisers] couldn't have got that more wrong.
"Jordan Henderson, don't forget, is in a similar contractual position, but the differences between their situations is enormous. Jordan has played as well, if not better, than at any time in his career this season.
"He will be out of contract in 12 months' time but, as negotiations continue, you are not hearing anyone calling him a money-grabber. Why? Because he and his agent aren't using the media to engineer a move or put pressure on Liverpool to improve his contract.
"This, more than anything, is my biggest problem with Sterling and his agent. For a 20-year-old and his agent to be taking on Liverpool FC in the public domain is a disgrace.
"I have no issue about players maximising their earning potential, but the agenda that is being driven from Sterling's camp is beyond a joke.
"The only way any good could have come from the BBC interview was if Sterling had said he was going to sign a new deal.
"Anything else, as he has discovered, was always going to create a massive problem. They cancelled talks with the club, but not with the media. If you don't want people talking about an issue, don't talk about it yourself."
Carragher believes Sterling would be unwise to leave Anfield at this stage of his career.
"What Sterling's camp are struggling to see is that Liverpool is the best place for him," he added.
"Forget money for a minute. This is about football and his development. He will start the vast majority of games at Anfield. Would that be the case at Arsenal? No.
"He wouldn't play as a centre-forward there ahead of Olivier Giroud or Alexis Sanchez. Being a number 10 would be out of the question, as he is not like Mesut Ozil or Santi Cazorla, which would mean a battle with Danny Welbeck and Theo Walcott to start wide right.
"If he went to Manchester City, he would be a squad player again. City play 4-4-2 under Manuel Pellegrini, which would mean him competing with Jesus Navas and Samir Nasri to start on the right.
"As for talk about Bayern Munich (Arjen Robben and Franck Ribery) and Real Madrid (Cristiano Ronaldo and Gareth Bale), he wouldn't get near either of those teams now.
"Liverpool and Rodgers are what Sterling needs at this stage of his career. He should realise this. There is one change I would recommend he makes this summer - and it is not his football club."
