The report of the Hillsborough Independent Panel on 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 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.
"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."