According to court documents, in 2004, Woods and his friend Kerry Spencer got into an argument with two of the police officers and threatened them.
Later that day, the officers were killed as they tried to arrest Woods and Spencer at a suspected drug house in the city of Birmingham. One other officer was injured.
Prosecutors said it was Spencer who had opened fire on the officers but accused Woods of luring the officers.
Woods was convicted by 10 out of 12 jurors. Alabama is the only US state that does not require a unanimous vote to enforce the death penalty.
Spencer was also handed the death penalty and is still on death row. In an open letter, he wrote: “Nathaniel Woods is 100% innocent, I know that to be a fact because I’m the person that shot and killed all three of the officers.
“Nathaniel Woods doesn’t even deserve to be incarcerated, much less executed.”
Kim Kardashian West, one of the high-profile figures who championed Wood’s case, said after the stay of execution was lifted: “My heart and prayers are with Nate and his family.”
Martin Luther King III, the son of Martin Luther King Jr, called Woods’ death a “mockery of justice”.