The QPR striker, who spent last season on loan at Newcastle, had been set to seal an £8.5million 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 Sky sources understand 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."
Speaking on Monday from QPR's Harlington training ground, Sky Sports News reporter Paul Gilmour explained: "The deal is off and having originally been disappointed and angered at the way the news was leaked, it's now our understanding that Liverpool and Queens Park Rangers have been in conversation.
"Phil Beard and Ian Ayre have had talks this morning. In the course of that conversation, Liverpool have told Queens Park Rangers the deal is off and have promised not to talk about Loic Remy any further in the media."
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.
"Luis Suarez isn't a Liverpool player so I don't need to talk about or reference Luis Suarez," Rodgers added to Liverpool club's official website.
"I'll just worry about the players I have, who showed in enough games last season and the season before, when we didn't have Luis, that we can score goals. That won't be an issue.
"I've seen a difference in Daniel since he came back; he has obviously got confidence from the World Cup and his season last year.
"I think he's ready to take on the mantle. He got 20-odd goals last season so he's a striker of great value.
"Markovic will only get better and better. He showed great strength when he burst through and hopefully later on when he picks up that speed and sharpness in his game, he'll go on and finish that.
"We've got good speed. Raheem Sterling is always a threat - he's a joy to watch. I thought those three were really exciting."