The veteran and charity fundraiser died in Bedford Hospital on Tuesday morning after testing positive for coronavirus.
Following his death, the former footballer shared a message alongside a video of his encounter with Sir Tom last year.
He described him as a “true hero” on Instagram, adding: “What he achieved for our NHS will never be forgotten.
“Tom, you were truly the very best of British.
“When our country and our world was struggling you gave us hope, put a smile on our faces and made us believe that tomorrow would be a good day. Rest in Peace Sir Tom.”
David met Sir Tom when he presented him with a framed football shirt after he became the first member of the Lionhearts squad of inspirational heroes.
In the video, filmed at Sir Tom’s Bedfordshire home, the former England captain said he knew he “had to dress up today”.
Sir Tom replied: “I put this on because it’s the only jacket I’ve got.”
Piers called Captain Tom a “magnificent man” and a “national hero”, writing: ”Let us all heed his mantra in our own lives: ‘Tomorrow will be a good day.’ Thank you, Tom.”
Piers’ Good Morning Britain co-host Susanna Reid also paid her respects, writing: “The man who united and inspired our nation at a moment of deepest despair. Thank you Captain Sir Tom Moore for your service.
“I am heartbroken for your family. A life well lived, and well loved.”