The report of the Hillsborough Independent Panel yesterday [Wednesday] revealed South Yorkshire Police changed statements relating to the 1989 tragedy, when 96 fans died, in order to blame supporters.
Prime Minister David Cameron yesterday [Wednesday] offered a "profound" apology to the families of those who died, telling the House of Commons that the report made clear "Liverpool fans were not the cause of the disaster".
The club and their fans have had to endure chants from rivals supporters about the events at Hillsborough, although some Liverpool fans are not entirely without blame in a similar situation, as songs about Manchester United's Munich air disaster have been sung in the past.
Rodgers hopes such chants will now cease in the light of what was made public yesterday [Wednesday].
"I speak as a human being and I don't ever like to hear anything like that, whatever club it is, that associates with other people's tragedies and death," said the Reds boss.
"Unfortunately you have a very small percentage of idiots at any club who will always try to smear another club's reputation.
"Of course, it is obvious these are chants that no-one wants to hear about any club.
"Unfortunately there are that minority of supporters who will maybe disappoint but let's hope we can all move on and we can all learn from this whole process."
Rodgers said the smear campaign which had been perpetrated against the fans was something which had gained a global foothold and he sympathised with what the families had faced up to.
"There is no doubt the propaganda at the time, if you were outside the city of Liverpool and its people, planted seeds throughout the country and the world," added the Northern Irishman, who was among a number of club representatives who attended a vigil in the city last night.
"It is the people who have suffered over all the years who have had to contend with that and fight against that sort of relentless campaigning at the time.
"But having come into the city and been around the people and being at the vigil last night there has never been any qualms about any sort of justification.
"We have all suffered death in our lives but to have it happen to your son, daughter, brother, sister, and then have a campaign against that - you can't begin to imagine what that must feel like."
Rodgers explained he had nothing but admiration for the work of the families of the bereaved who fought for the truth to be revealed.
"It has been incredible what they have had to go through," he said.
"I was a 16-year-old boy in Northern Ireland at the time and I sit here today as a 39-year-old and the families have just had justice.
"It is absolutely incredible it has taken all that time.
"My respect goes out to everyone involved in that process over 23 long, hard, arduous years and hopefully they found some sort of justice yesterday.
"I have met with some of the campaigners since I've been here and had a shortened version of what they have been through and it has been remarkable really for all the families.
"The inspiring effect it has had on me alone, and certainly for the club, is that you fight: there is a cause there and if you feel strongly enough about it you must be relentless to make a wrong right.
"These people have done that. They have suffered - none of us can begin to imagine what they have been through.
"Everyone will draw inspiration from how hard they have fought and the time and effort which goes into protecting those loved ones.
"I don't think it's only a message for the football club and the city, I think it is a world message."