The 74-year-old is accused of killing his wife and childhood sweetheart Janice, who he was with for 56 years.
The pair, who are grandparents, retired to Paphos in Cyprus 20 years ago.
Janice, 75, had blood cancer and was found dead at the home they shared in the village of Tremithousa in Paphos on December 18 last year. Her husband was found heavily sedated and spent two weeks in a coma. Once he came round, police charged him with murder.
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His trial was due to start today (Thursday), with defence lawyers arguing that Hunter should instead be charged with assisting a suicide.
State prosecutor Andreas Hadjikyrou said the postponement was the result of a court scheduling conflict and would resume on September 9.
Hunter will remain in custody until then.
He is being represented by Justice Abroad, a group which defends Britons embroiled in legal difficulties in foreign countries.
Its director Michael Polak said: “Although David is disappointed that the trial did not start today, he is determined to fight this case and we will be challenging the evidence at every stage in our mission to bring David home.
"David is 75 years old, and it is difficult to see any public interest in prosecuting him for murder. He would like to thank all of those from around the world how have sent him messages of support at this difficult time.”
It will be the first euthanasia case to be tried on the east Mediterranean island and comes as lawmakers debate whether to decriminalise euthanasia, amid strong opposition from conservative circles, including the influential Orthodox Church.
Cyprus’ attorney general George Savvides rejected a defence request to reduce the charge to assisted suicide, which would likely keep Hunter out of jail. If he is tried for murder and convicted, he is likely to die in prison.
Prosecutors claim there is no evidence, such as a note, to suggest Mrs Hunter had ever asked her husband to help her die.
They also dispute there was any medical diagnosis proving that she suffered from blood cancer.
The family is raising funds to assist with the legal costs. To donate, go to https://www.crowdjustice.com/case/help-bring-david-home/