Amnesty International said on Monday that Scott Panetti, who is due to be killed in Texas on 5 February, should be pardoned because of his history of mental illness.
Panetti, 45, shot dead his parents-in-law in 1992 and then kidnapped his wife and daughter, holding them at gunpoint.
But Amnesty says he is a diagnosed schizophrenic and there is evidence he was psychotic at the time of the killings.
The rights group added that Panneti was unfit to stand trial for his crimes.
At his trial the Texan acted as his own lawyer, which he did dressed as a cowboy and subpoenaing John F. Kennedy and Jesus Christ. At the end of the trail, jurors voted for death.
"His (Panneti's) trial has been variously described as a circus, a farce, immoral and a mockery of justice by people who were there. One word for his impending execution is 'indecent'... If (the) verdict is carried out, it would be one more shameful episode in the ugly history of state killing in the USA"
However, three years later, Panetti's former wife filed a petition saying he was suffering from paranoid delusions at the time of the killings. She added her parents tried to get the police to remove his guns shortly before the crime.
"His trial has been variously described as a circus, a farce, immoral and a mockery of justice by people who were there", Amnesty International said. "One word for his impending execution is 'indecent'." The impending execution of this mentally ill man once again shows the arbitrariness and unfairness of capital justice in the USA."
"Some of the jurors indicated that they voted for execution out of fear. If their verdict is carried out, it would be one more shameful episode in the ugly history of state killing in the USA."
Amnesty added: "The death penalty assumes absolute, 100% culpability.
Amnesty's call comes after the European Union sent a letter to Texas Governor Rick Perry last month, appealing for Panneti's life.
Texas execution rate
"The European Union opposes the death penalty in all cases and accordingly aims at its universal abolition, seeking a global moratorium on the death penalty as a first step," the letter said.
"The European Union considers that in countries which have not yet abolished the death penalty, this penalty should not be imposed on persons suffering from a mental disorder."
THE DEATH PENALTY AND THE US
- 891 people have been executed since 1977
- 38 of the 50 US states provide for the death penalty in law
-The death penalty is also provided under US federal military and civilian law
Texas has the highest execution rate of all America's states.
Multiple murder is a capital offence in Texas, and opinion polls show that most US citizens support the ultimate penalty for such crimes.
Next month, the USA may see its 900th execution since resuming judicial killing in 1977. The total is currently 891.
However, the US Supreme Court has ruled that the mentally ill should not be executed.