The Football Association issued the suspension and a £100,000 fine after investigating the midfielder's comments made towards Wolves' Morocco international Romain Saiss in September.
The FA found Shelvey guilty of using "abusive and/or insulting words" that "included reference to ethnic origin and/or race and/or nationality".
The 24-year-old says he will not appeal against the ban but insists he did not use the alleged language.
"I am very disappointed and frustrated with the outcome of the hearing. I strongly maintain that I did not use the offensive language that has been alleged," he said in a statement released by the club.
"Despite my strong disappointment, we think an appeal is very unlikely to change the panel's decision on the case, so I have decided not to take that option.
"Appealing the ban would also extend the period of uncertainty for the club and the team during such an important part of the season.
"This would not be fair on my club, manager or team-mates and I want to draw a line under this now so that I can return to help my team with our promotion challenge as soon as possible."
Manager Rafael Benitez said: "Jonjo has maintained his position from the outset and as such has received our full support. Despite this issue going on in the background he has continued to train and play really well.
"Now he has made the decision not to appeal the verdict we can focus on the upcoming games without uncertainty and distraction."