Shelvey, 24, was charged over an incident involving Wolves' Moroccan midfielder Romain Saiss in September.
"I strongly maintain that I did not use the offensive language that has been alleged," said Shelvey.
The Englishman was also fined £100,000 and ordered to attend a Football Association education course.
Shelvey added: "I am very disappointed and frustrated with the outcome of the hearing.
"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."
Newcastle 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."
It is understood the FA - which also had a right of appeal against the independent panel's sanction if it felt it was inappropriate - will not be contesting the ban either.
Newcastle were losing to Wolves in their Championship match at St James' Park on 17 September, when an incident occurred between Shelvey and Saiss in the 87th minute.
In charging the former Liverpool and Swansea player, the FA said his misdemeanour was an "aggravated breach" of rules, and that it "included reference to ethnic origin and/or race and/or nationality".
BBC Sport's Pat Murphy said the matter was brought to referee Tim Robinson's attention by one of Saiss' team-mates immediately after the game.
FA guidelines, which were revised in 2013, state any player found guilty of racially abusing an opponent or discriminating on religion, sexuality or disability, should be banned for at least five games.
Prior to that, Chelsea's John Terry was handed a four-match ban and a £220,000 fine for racially abusing QPR defender Anton Ferdinand, while Liverpool's Luis Suarez received an eight-match suspension and a £40,000 fine for his abuse of Manchester United's Patrice Evra - both in 2011.
Newcastle United midfielder Jonjo Shelvey will not contest a five-game ban given after he was found guilty of using racially abusive language.