Paya and another Cuban dissident, Harold Cepero, died on July 22, 2012, in southeastern Cuba when their rental car went off an unpaved country road and crashed into a tree.
The vehicle's driver, Angel Carromero, a leader of the youth wing of Spain's ruling Popular Party, and another passenger, Swedish activist Jens Aron Modig, survived the crash.
An official Cuban enquiry found that Carromero had been driving above the speed limit at the time of the accident, causing him to lose control of the vehicle.
He was convicted in Cuba of vehicular homicide and returned to Spain to serve out a four-year sentence.
But in an interview published in Spanish newspaper El Mundo, Carromero said the official version that he was speeding was "a perfect alibi to hide the death of the only dissident who could lead the transition in Cuba".
"I am convinced that he survived the accident. The nurses and a priest told me that all four of us were admitted to the hospital," he added.
"Cuban secret services assassinated Oswaldo Paya."
Carromero blasted his trial in Havana as a "pantomime" and accused Cuban prosecutors of "fabricating the evidence".
Paya's family have always said they suspected foul play and accused the Cuban authorities of involvement in his death. They are seeking an international investigation.
Spain's opposition socialist PSOE party responded to the El Mundo interview on Monday by demanding to know what the Spanish Government's official posture towards Cuba would now be.
"In this case, we have been very responsible and supported the Government in all of its actions so that Carromero could return to Spain," said the PSOE's Secretary for Institutional Relations and and Regional Relations, Antonio Hernando.
"Now, after these grave accusation, we demand that the Government take action," the official said.
In an interview with the Washington Post in March, Carromero had already said that the car was struck from behind by another vehicle just before the crash.
Carromero has been equipped with an electronic bracelet and is serving out his sentence at home. He works as a consultant at Madrid's town hall, which is run by the Popular Party.
Paya was the lead organizer of the Varela Project, a signature-gathering drive asking authorities for a referendum on guaranteeing rights such as freedom of speech and assembly in Cuba.