Yang was handed a suspended death sentence earlier this month after a court found him guilty of opaque "espionage" charges, which the Australian citizen has forcefully denied.
The sentence has been suspended by two years and there is a reasonable chance it could be commuted to life in prison if officials are pleased with Yang's behaviour.
The father-of-two, whose health has deteriorated as a cyst grows on his kidney, told his family he no longer had the strength to fight the charges.
"Commencing an appeal would only delay the possibility of adequate and supervised medical care, after five years of inhumane treatment and abject medical neglect," his family said in a statement.
"Yang's deteriorating physical condition does not allow him to endure further rounds of this legal system," the statement added.
Instead, Yang's family said they would focus on securing urgent medical relief for his "serious kidney condition".
Chinese foreign ministry spokesperson Mao Ning on Wednesday defended Beijing's treatment of Yang, saying the country's justice system would "continue in accordance with the law to protect the legal rights and interests of the party involved".
"China is a country with rule of law, and China's judicial organs will handle the relevant case in accordance with the law," Mao said at a regular press conference.
The unexpected sentence has sent a chill through Australia-China relations, which have been on the mend after years of bickering and trade reprisals.
"We will continue to press for Dr Yang's interests and wellbeing, and provide consular assistance to him," Foreign Minister Penny Wong said in a statement Wednesday.
"I acknowledge the strength that Dr Yang's family and friends have demonstrated through this period.
"All Australians want to see Dr Yang reunited with his loved ones."
The Chinese-born Australian has been detained in China since 2019, accused of spying in a closed trial heavily criticised by human rights activists.
The writer and academic, who also goes by the pen name Yang Hengjun, had previously told supporters he was tortured at a secret detention site and feared forced confessions may be used against him.
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