"I've taken drugs but have never been addicted to them. You couldn't move in Hollywood in the 1980s [when he worked there] without seeing bowls of cocaine everywhere. But I don't do drugs and I barely drink."
He also denied reports he did not attend Meghan's 2011 wedding to Hollywood director Trevor Engelson. The couple divorced just two years later.
Published photographs from the four-day celebration in the luxury resort of Ocho Rios, Jamaica, show the couple exchanging vows on the beach and Meghan's mother, Doria Ragland, paddling in the sea. But none shows Mr Markle, who was divorced from Doria by then.
"There are no images of me frolicking on the beach because, at the time, I was well into my 60s and not much into frolicking," he said. "But I was very much there and there were more than 100 people there to witness it. Doria and I walked Meghan down the beach together.
"Of course I have photographs of Meghan and me at her first wedding but I choose not to share them because I think that would be in poor taste and offensive to Prince Harry. We booked out the whole resort and I was more than happy to contribute to the cost of my daughter's first wedding.
"I sold a bunch of Facebook shares at a low price to help pay for my share of it. I don't regret it in the slightest because I was happy to help out, but that stock would be worth a fortune now. I'm not broke, I'm not bitter and I never expected Meghan to pay me back for anything. I was happy as a father to give her a great education. I was happy to provide the best start in life to all my children. Of course I never expected her to pay me back for her schooling."
Mr Markle has not spoken to his daughter since Prince Harry called him in hospital after his heart attack days before the Royal Wedding on May 19. During the call, Harry scolded him for co-operating with paparazzi over staged pictures, including one of him apparently being measured for a wedding suit.
"I was in a hospital bed, I'd just had a heart attack. The timing wasn't great," he said.
"Prince Harry said, 'If you'd listened to me none of this would have happened', meaning I shouldn't have co-operated with the paparazzi. And he's completely right. I made a terrible mistake which I have paid for ever since.
"There was no yelling or screaming and I didn't slam the phone down. How do you slam a cellphone down?" As the MoS reported last week, Mr Markle still hopes for a reconciliation with Meghan. He said: "I love my daughter. I love Prince Harry. I wish them nothing but the best."
If there is to be a face-to-face rapprochement with the couple, it might come in February.
Sources in the US have suggested that "early talks" have begun about Harry and Meghan attending a charity polo match in Los Angeles or Santa Barbara, California, as part of a US tour.