The QPR striker, who spent last season on loan at Newcastle, had been set to seal an £8.5million ($14.5million) move to Anfield after agreeing personal terms.
A delay in confirming the France international as the Reds' fifth summer signing was initially attributed to administrative issues but it is understood Liverpool pulled the plug on medical grounds.
Speaking after his side's 1-0 International Champions Cup win over Olympiakos in Chicago, Rodgers said: "It's very simple, we have decided not to go ahead with the deal.
"It's unfortunate for the player - we're obviously disappointed - but there's nothing more to be said. We move onto other targets."
Remy had been offered the number 7 shirt vacated by Luis Suarez but after watching England team-mates Raheem Sterling and Daniel Sturridge operate in an attacking three alongside new arrival Lazar Markovic, Rodgers remains confident his side already possesses ample firepower.