Thomas Markle (73) said he would no longer be flying to Britain after embarrassing disclosures about his dealings with the paparazzi.
Mr Markle reportedly told the US showbusiness website TMZ that he had suffered a heart attack six days ago, but had checked himself out of hospital to travel for the ceremony on Saturday.
The decision leaves arrangements for the royal wedding in doubt, and Ms Markle without her father on the biggest day of her life.
It also raises the question that her mother, Doria Ragland, may walk her down the aisle, or that she will be accompanied by a member of the royal family or make the journey alone. Kensington Palace issued a statement asking for understanding.
A spokesman said: "This is a deeply personal moment for Ms Markle in the days before her wedding. She and Prince Harry ask again for understanding and respect to be extended to Mr Markle in this difficult situation."
Ms Markle is understood to be deeply worried about her father, who has been struggling to cope with being in the spotlight as a result of her engagement.
Palace aides, who have been attempting to help Mr Markle in dealing with media interest, have expressed concern about the pressure he has been under.
Mr Markle was last night said to be "devastated", with TMZ saying he had spoken to his daughter to apologise.
On May 4, a palace spokesman confirmed that Mr Markle intended to walk his daughter down the aisle, after months of speculation.
He and Ms Ragland were also scheduled to meet the queen, Prince Philip, Prince Charles, his wife Camilla, and Prince William and his wife Kate, in a show of support from the royal family.
Ms Markle has been beset by difficulties in the lead-up to her May 19 wedding, with her estranged half-sister and half-brother regularly criticising her.
Her father, who lives alone in Mexico, was last weekend revealed to have collaborated with the paparazzi for a series of photographs.
The disclosure was particularly embarrassing after Kensington Palace had issued repeated warnings to the media not to publish pictures of Mr Markle, insisting he had no wish to take part in any personal publicity.
The photographs aroused suspicion after showing Mr Markle exercising in public, reading a book called 'Images of Britain', being measured for a suit, and looking up pictures of his daughter in an internet café.
Last weekend, 'The Mail on Sunday' published CCTV images showing him collaborating with a photographer, walking into the cafe with him and setting up the best angle.
TMZ quoted Mr Markle as saying he had meant no harm to his daughter or the prince, and had not made the decision for money but in order to recast his image after a series of unflattering photographs were taken without his co-operation.
"He's now decided not to go [to the wedding] because he doesn't want to embarrass the royal family or his daughter," TMZ reported.
Hours earlier, Ms Markle's half-sister claimed responsibility for setting up the paparazzi photo shoot for her father.
Samantha Grant, who is no longer in contact with Ms Markle, said she had suggested their father pose for "positive" photographs.
Pressed on whether he had been paid for his co-operation during a television appearance, Ms Grant said: "I have no idea, but if he did I'm going to assume it was a pittance."
Mr Markle has been receiving the support of Kensington Palace since news of his daughter's engagement brought him into the public eye, with a source saying that aides would "continue to make interventions with the media out of concern for his safety and security". (© Daily Telegraph, London)