Paul Wright, 34, was on holiday in Crete in 2003 when he said he was a passenger in a car which crashed.
He was arrested in March after being convicted in his absence by a Greek court of joyriding and criminal damage.
Mr Wright said he was "really happy" with the CPS's decision.
The father-of-two, from Mold, had faced either a Â£3,500 fine or a 15-month prison sentence in Greece.
But at a hearing to fight the extradition order last week a judge ruled in his favour, saying he was not satisfied Mr Wright had deliberately absented himself from the Greek proceedings.
The CPS had seven days in which to appeal but told Mr Wright's solicitor on Tuesday that it would not be pursuing the case.
"I feel better now I have the finality of it because when I got the decision last week I was still left in limbo," said Mr Wright.
"More than anything I can actually start planning to build my life again because I had so much taken away from me in the last year - my health, my career and then in March my liberty and the potential of my family life.
"Everything was just stripped away and now I've just been handed it all back."
Mr Wright, who cannot work because of a serious back injury, estimates his case has cost taxpayers about Â£30,000 in legal fees.
He claims he is innocent of the charges and that his friend crashed the car - which belonged to a Greek barman - ran off and left him to explain the damage.
His case was supported by Delyn MP David Hanson, who said it was unfair for a person to be convicted in their absence with no opportunity to defend themselves.
Mr Wright said Tuesday's news was "just round one" in the fight to clear his name.
"What I have won today is the right to stay in the UK. If I ever want to travel again, I have to now fight it in Europe.
"I've got to clear my name. I've been a prisoner in my own home since March because of my bail conditions and now I'm a prisoner in my own country and I can't accept that."
A Flintshire man who won his fight against extradition to Greece for a crime he said he did not commit has learned the Crown Prosecution Service will not be appealing the decision.