"I won't be saying anything on Luis Suarez and it's only purely out of respect for the players that are working hard at the minute," Rodgers said after Liverpool's 1-0 win at home to Stoke City in their Premier League opener on Saturday.
"All I will say is I am very satisfied with how everything has been resolved and we as a club are now looking to the future and moving on," he told a post-match news conference.
Suarez returned to first-team training on Friday and attended the match at Anfield with his family, receiving a generally warm reception from the home fans.
He is midway through a 10-match ban for biting Chelsea's Branislav Ivanovic towards the end of last season.
"He's a part of our team and a part of the club," Rodgers added. "The supporters are a class act here - the classic Liverpool Way is to always support the players and the manager. That's what they always do."
Rodgers had previously said Suarez, who scored 30 goals in all competitions for Liverpool last season, had to apologise for his public outburst if he was to return to first-team training.
The Uruguayan was the subject of two bids from Arsenal, including an offer of 40 million pounds ($62.45 million) and one penny which he believed would trigger a release clause, something that Liverpool have denied.
Liverpool striker Daniel Sturridge scored the only goal against Stoke just before halftime and new goalkeeper Simon Mignolet saved a penalty in the 87th minute from Jonathan Walters.
(Reporting by Josh Reich in London, editing by Ken Ferris)
