Liverpool striker Sturridge scored twice and missed a penalty in a 4-0 thrashing of Everton at Anfield on Tuesday.
The England international was then brought off just after wasting another chance to complete his hat-trick and was clearly unimpressed with Rodgers' decision.
"I apologise for my reaction," he said on BT Sport.
"I was disappointed by missing the penalty and felt I'd let myself down as well as the fans.
"The hat-trick would have been the icing on the cake and it's unfortunate.
"I was disappointed with myself. It was nothing to do with the manager and I'd like to apologise for my reaction if anybody took any offence. It’s not about me, it’s about the boys and a great victory."
"In fairness to Daniel, he was frustrated after not scoring the penalty," Rodgers said.
"We are very much a team and if anyone is unhappy we say to just keep it to yourself and speak to me the next day, but there is no problem with Daniel. He was brilliant tonight."
Roberto Martinez fancied Everton's chances of winning at Anfield for the first time since 1999 ahead of the game, but there was a gulf in class on Tuesday as Liverpool swept their neighbours aside.
The Liverpool manager said: "It was a gamble to go with two strikers but that's the reason we took the risk. We got four goals and could've had six or seven.
"It was a brilliant demonstration of how to play under pressure."
