David Gale is a professor who is a prisoner on death row in Texas. With only a few days until his execution, his lawyer negotiates a half million-dollar fee to tell his story to Bitsey Bloom, a journalist from a major news magazine. She is known for her ability to keep secrets and protect her sources. He tells her the story of how he ended up on death row, revealed to the audience through a series of lengthy flashbacks.
Gale is head of the philosophy department at the University of Texas and an active member of DeathWatch, an advocacy group campaigning against capital punishment. At a graduation party, he encounters Berlin, an attractive graduate student who had been expelled from the school. while Gale is drunk at a party, she seduces him and gets him to have rough sex with her. She then falsely accuses Gale of rape. The next day, he loses a televised debate with the Governor of Texas when he is unable to point to an example of a demonstrably innocent man being executed during that governor's term. After losing the debate, Gale is arrested and charged with rape. While the rape charge against Gale is later dropped, the damage had already been done, and his family, marriage, university career and reputation are all destroyed.
Constance Harraway, a fellow DeathWatch activist, is a close friend of Gale who consoles him after his life falls apart, and the pair have sex. However, the next day, Harraway is discovered raped and murdered, suffocated by a plastic bag taped over her head. An autopsy reveals that she had been forced to swallow the key to the handcuffs used to restrain her, a psychological torture technique used under the communist regime of Nicolae CeauĹescu, which Gale and Harraway had both protested against. The physical evidence at the crime scene points to Gale, who is convicted of her rape and murder and is sentenced to death.
In the present, Bloom investigates the case in between her visits with Gale. She comes to believe that the apparent evidence against Gale does not add up. She is tailed several times in her car by a person who turns out to be Dusty Wright, the alleged one-time lover and colleague of Harraway, who she suspects was the real killer. Wright slips evidence to Bloom that suggests Gale has been "framed", implying that the actual murderer videotaped the crime. Bloom pursues this lead until she finds a tape revealing that Harraway, who was suffering from terminal leukemia, had committed an elaborate suicide made to look like murder. She and Wright are both seen on the videotape, showing that they framed Gale as part of a plan to discredit the death penalty (by showing that an innocent person can be executed).
Bloom does not find this evidence until the day of Gale's scheduled execution. She tries to give the tape to the authorities in time to stop the execution. She arrives at the prison just as the warden announces that it has already been carried out. The tape is then released, causing a media and political uproar over the execution of an innocent man. Later, Wright receives the fee that Bloom's magazine agreed to pay for the interview, and delivers it to Gale's ex-wife in Spain, along with a postcard from Berlin in San Francisco apologizing for the false rape accusation. His ex-wife looks distraught, knowing Gale told the truth and that she effectively stole their child away from him.
Much later still, a videotape labelled "Off the Record" is delivered to Bloom. This tape picks up at the point where Wright confirmed that Harraway was dead, then continues on to show him stepping aside to allow Gale, also present and party to the suicide, to caress her body. It was in doing this that Gale left his fingerprints on Harraway's plastic suffocation bag, showing he knowingly sacrificed himself to prove that innocent prisoners can be executed.

Question: Why does David lose the debate with the governor of Texas?
Answer: He is unable to point to a definite innocent person being executed during the governor's term.
Alternative answer: because he can't think of a single innocent person sentenced to death