The Russian said in a statement he was resigning "with immediate effect" from his post at the Moscow theatre as well as his equivalent position at France's Orchestre National du Capitole de Toulouse.
Sokhiev was appointed by the Bolshoi in 2014. He was brought in as part of moves to improve the theatre's image after scandals including the 2013 acid attack on its then-artistic director Sergei Filin.
He comes from the same North Ossetia region of Russia as star conductor Valery Gergiev and is considered to be his protege.
Kremlin loyalist Gergiev has been stripped of his role at the Munich Philharmonic for failing to denounce Russia's actions.
Sokhiev said in a lengthy statement that "many people were waiting for me to express myself and to hear from me my position on what's happening at the moment," referring to Russia's military action in Ukraine.
He said he decided to resign after "being forced to face the impossible option of choosing between my beloved Russian and beloved French musicians".
He cited opposition from authorities in Toulouse to his planned staging of a Franco-Russian music festival there, saying they "want me to express myself for peace".
Sokhiev became music director of the Toulouse orchestra in 2008 and continued to work with the orchestra after joining the Bolshoi.
Sokhiev did not say explicitly whether he backs or opposes Russia's actions in Ukraine while he said that "I have never supported and I will always be against any conflicts in any shape and form".
He said musicians are becoming "victims of so called 'cancel culture'" and suggested Russian music could come under threat.
"I will be soon asked to choose between Tchaikovsky, Stravinsky, Shostakovich and Beethoven, Brahms, Debussy," he warned.
Bolshoi general director Vladimir Urin told TASS state news agency he was saddened by Sokhiev's decision.
"I'm very sorry. His departure is a serious problem for the Bolshoi Theatre. It's unclear how the situation will develop from now."