Cook, 21, who had planned to appeal after Lutalo Muhammad was selected in the -80kg weight division, said in a statement on Friday that he had also declined an offer to be first reserve for the London Games.
"After careful consideration over the last few weeks, and despite my representatives advising me that I still had a good chance of successfully overturning the BOA's ratification of Great Britain Taekwondo's decision not to select me for London 2012, I have decided not to take my case against the BOA to the High Court," Cook said.
"The financial implications for me, and, more importantly, for my parents, of doing so are too prohibitive.
"In the best interests of Team GB, I have decided not to take up the offer to be first reserve for London 2012," added the statement from the -80kg European champion and world number one in the category.
"I wish all of the athletes the very best of luck - that includes Lutalo Muhammad.
"The Olympics, and especially a home Olympics, would have been the pinnacle of my career. I feel totally devastated."
British Olympic Association chief executive Andy Hunt said last week that Cook's lawyers had written asking the BOA to consider either re-visiting the ratification of the nomination of Muhammad or approaching the World Taekwondo Federation to request a wildcard.
Cook was left out of the British team after Muhammad, European champion in a heavier non-Olympic weight category, was preferred.
The decision triggered accusations that the selection was politically motivated and a punishment for Cook, who left British Taekwondo's performance programme last year to train separately.
The WTF has said it was reviewing the case.
The Olympics start on July 27 and the taekwondo events begin on August 8.