Piara Powar, executive director of European football's anti-discrimination body FARE, said Liverpool manager Kenny Dalglish's comments had been "undignified" and that their reaction had damaged the club's brand across the world.
Suarez is not appealing against the ban imposed for racially abusing Manchester United's Patrice Evra.
Powar told Press Association Sport: "This is a lack of respect for the governing body by Liverpool and the FA should charge Liverpool FC and Kenny Dalglish.
"I think the FA should come back now and be very clear that Liverpool could be construed to have brought the game into disrepute by the way in which they have consistently undermined the judgment and by Kenny Dalglish's comments.
"Liverpool have been too keen to support their man and in doing so have whipped up a sense of paranoia amongst their fans.
"This is not the Liverpool FC that we have applauded in the past for their support for a whole range of issues.
"The responses from Kenny Dalglish have been undignified; the way in which they have dealt with the whole matter has been unprofessional.
"For the club to so aggressively militate against what looks to most people a considered judgment from the FA leads to a potential for anarchy."
Powar said reaction from his colleagues in other parts of the world made him believe the affair was damaging Liverpool.
He added: "They have damaged their brand, no question.
"There is no question that Liverpool do have a global appeal, but I have emails from colleagues in Africa asking me what the hell is going on.
"I think people will be watching this and I believe there is no question that their plans for global expansion will have been damaged by this.
"That's not to say they cannot come back from this but it has done them a lot of damage and they have not conducted themselves in a very palatable way."
Powar's call comes after Lord Ouseley, the chairman on British football's anti-racism group Kick It Out, called Liverpool "hypocritical" and Suarez's apology "lamentable".
Ouseley also described Liverpool players wearing T-shirts in support of Suarez as "dreadful".
Suarez's apology stated: "I admitted to the commission that I said a word in Spanish once and only once. I told the panel members that I will not use it again on a football pitch in England. I never, ever used this word in a derogatory way and if it offends anyone then I want to apologise for that."
Suarez did not mention Evra by name in his apology, and previously said in a statement on Tuesday he would carry out the suspension "with the resignation of someone who hasn't done anything wrong'' saying the word 'Negro' was in his country a commonly-used word which did not show any lack of respect.
Ouseley, who was chairman of the Commission for Racial Equality from 1993 to 2000, wrote in The Guardian: "Liverpool FC need to take a hard look at themselves. Since the publication of the 115-page report of the findings of the FA's independent commission, Liverpool's vitriol has increased.
"Suarez's attempt at a belated apology is nothing short of lamentable. I cannot believe that a club of Liverpool's stature, and with how it has previously led on matters of social injustice and inequality, can allow its integrity and credibility to be debased by such crass and ill-considered responses."
Ouseley added: "Liverpool have been particularly hypocritical. You can't on the one hand wear a Kick It Out T-shirt in a week of campaigning against racism when this is also happening on the pitch: it's the height of hypocrisy.
"Liverpool players wore a T-shirt saying: 'We support Luis Suarez', seemingly whatever the outcome. This was a dreadful knee-jerk reaction because it stirs things up."