The 25-year-old was signed as a £35m replacement for Fernando Torres in January 2011, but had an injury-disrupted first 18 months on Merseyside.
He returned from European Championship duty with England in 2012 in fine spirits, but after only 18 minutes of game time under new Reds boss Rodgers was allowed to join the Hammers on loan.
He has since moved to Upton Park on a permanent deal and, speaking ahead of title-chasing Liverpool's visit on Sunday, the former Newcastle United forward has complained that he was not given enough time to show what he was capable of under the Rodgers regime.
He told Sky Sports News on his time at Anfield: "Obviously it wasn't what I wanted. I went there and I wanted to kick on.
"I had problems with injuries and was playing one game out of three and never really got going. If I did get going, my career at Liverpool, I would still be there now. It didn't, now I'm here and I'm happy."
When asked if he deserved more time in which to impress a new manager, Carroll added: "I think so. I was away with England at the time and I felt that I was good and confident about coming back to Liverpool and showing what I'm about.
"I never really got the chance and was told to leave. That's why I'm here."
Carroll scored 11 goals in 58 appearances for Liverpool, who are two points clear at the top of the table.