The subject of two failed bids from Arsenal, Suarez was forced to train on his own by Rodgers after declaring that he wanted to leave, but the 26-year-old Uruguayan returned to training with his team-mates on Friday.
Speaking after Liverpool's 1-0 win over Stoke City in the opening game of the Premier League season, Rodgers said he felt the issue had been put to bed.
"I am satisfied with how everything has been resolved and we as a club are looking to the future," Rodgers told BT Sport.
"There is nothing to be said there. I won't be saying anything on Luis Suarez and that is only purely out of respect for the (other) players."
He briefly appeared beside the pitch prior to the game against Stoke, carrying his daughter, Delfina, and received a short burst of applause from the fans already inside the stadium.
"He is a part of our team and club and the supporters are a class act here," said Rodgers.
Speaking ahead of the game, Rodgers had confirmed that Suarez was "back in the group", but he refused to confirm reports that the player had apologised to his team-mates for his conduct.
Liverpool were ultimately indebted to new goalkeeper Simon Mignolet for saving an 89th-minute penalty from Jon Walters, after Daniel Sturridge had given the hosts a 37th-minute lead with a crisp shot from 20 yards.
However, despite the narrow winning margin, Liverpool's forward line of Sturridge, Philippe Coutinho and debutant Iago Aspas played with plenty of fluidity and invention, and Rodgers was thrilled by their contribution.
"It's always a concern as a manager when you dominate the game so much and create so many chances, but today we only scored one," said the former Swansea City manager.
"I was so, so pleased for the players because they have worked tirelessly and it starts our actual season on three points. The fluidity and flow of our game was good, our passing and combination play was excellent. I thought our intensity from the off was outstanding."
Rodgers also paid tribute to Mignolet, a Â£9 million ($13.9 million, 10.6 million euros) signing from Sunderland, whose arrival saw previous first-choice goalkeeper Pepe Reina loaned to Napoli.
"It was a terrific performance all-round, but of course the goalkeeper gets the plaudits when he saves a penalty at the end," said Rodgers.
"He had done his homework. He made a good first save and his second action to get up and make the double save (from Kenwyne Jones's follow-up) was impressive.
"From that, you also saw the spirit in the team and what we are trying to cultivate here."
Stoke goalkeeper Asmir Begovic was named the official man of the match for largely keeping Liverpool at bay, but he was disappointed to taste defeat in new manager Mark Hughes' first game in charge.
"It doesn't mean a lot when you can't get a result," said the Bosnian international. "We put in a decent performance and should have got something out of the game.
"The spirit was there and we kept digging in and we caused a lot of problems in the end. We are being positive to take stuff out of the game and kick on next week."
Stoke assistant manager Mark Bowen, standing in for Hughes after the Welshman departed on a scouting mission, said Walters had not let anybody down by failing to convert his penalty.
"Jon Walters gives everything for this club, so nobody in the dressing room will do anything other than pick him up and get him ready for the next game," he said.
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