Drama over whether Markle would attend threatened to overshadow the wedding of the 36-year-old American actress to Harry (32) at Windsor last month.
The 73-year-old had been due to take part in the ceremony at St George's Chapel, but after a flurry of sometimes contradictory statements to an American news website, he eventually stayed away, saying he needed heart surgery.
In the end, Meghan walked down the first part of the aisle alone, before she was joined by Harry's father, Prince Charles, the heir to the throne.
Markle, a former lighting director for TV soaps and sitcoms, told ITV's Good Morning Britain that he was sad to have missed the ceremony, but also proud of his daughter and grateful to the Queen's oldest son for stepping in.
"Watching it was difficult for me, because I wasn't there, but at the same time I'm watching my daughter being watched by the entire world. Everybody was watching my daughter. I couldn't have been more proud," he said.
"How could I have asked for a better replacement than Charles?... I just wish it had been my hand holding my daughter, not his. But he was wonderful to them."
Markle also apologised for staging photos with a photographer which contributed to the furore around him ahead of the wedding.
"I thought that would be a nice way of improving my look. Well obviously that all went to hell," he said.
Meghan's parents are divorced and while mother Doria Ragland (61), has met Harry and attended the wedding, her father has yet to meet the prince.
However, Markle said that Harry had rung him to ask permission to marry Meghan, former star of the TV drama Suits, and that he was very likeable.
"He's great, an interesting guy, a prince, but my daughter’s been a princess since the day she was born," he said.
Markle said that as well as Harry, he would one day like to meet the Queen.
"I have had respect for that woman since I was a child," he said. "She is one of the most incredible women in the world and I would love to meet her."
