The 67-year-old ex-Trump adviser was convicted in 2019 of lying to Congress, obstruction and witness tampering.
He had been due to report to a federal prison in Georgia on 14 July.
Speaking to Fox News host Sean Hannity on Monday night, Stone said he believed his reprieve had saved him from “a near-death sentence”.
“I was literally hours away from being sent to a Covid-infested prison in violation of the current Bureau of Prisons and DoJ [Department of Justice] policies,” he said in his first TV appearance since Friday’s presidential clemency.
He added: “I’m 67 years old, I’ve had a lifelong problem with respiratory problems.”
Earlier on Monday, Mr Trump told reporters at the White House: “I am getting rave reviews for what I did for Roger Stone.”
The charges against Stone were linked to Special Counsel Robert Mueller’s investigation into alleged collusion between Russia and the 2016 Trump campaign.