The former VfB Stuttgart and Bayern Munich forward scored 26 league goals last season as he helped the club win the Turkish league, a performance which also earned him a place in the Germany squad for Euro 2016.
"The reason for this is completely down to the political situation," the 31-year-old said on his Facebook page.
"There are no sporting or any other reasons that have made me make this decision. It is purely down to the terrible events that happened in the last few days," added Gomez, who scored twice at Euro 2016 as he helped Germany reach the semi-finals.
About 60,000 soldiers, police, judges, civil servants and teachers have been suspended, detained or are under investigation since a military coup attempt on Friday.
The failed putsch and the purge that followed it have both unsettled the country of 80 million, which borders Syria's chaos and is a Western ally against Islamic State.
Gomez, who missed the 2014 World Cup due to persistent injury problems, played on loan from Italian club Fiorentina last season and Besiktas had been hoping to sign him on a permanent basis.
"I have to tell you Besiktas fans personally that it is very heavy for me that I will not be playing for this great club in front of you wonderful fans in this unique stadium anymore," Gomez said.
"I hope that these political problems can be solved in a peaceful manner soon. Then it would be my great wish to play for Besiktas again."
(Writing by Brian Homewood; Editing by Toby Davis)
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