Mr Markle told celebrity news site TMZ he would no longer be walking her down the aisle, following reports in the Mail on Sunday he staged photographs.
Samantha Markle told Good Morning Britain he had suffered a heart attack due to stress caused by the media.
Kensington Palace has said it is "a deeply personal moment for Ms Markle".
The widely circulated paparazzi pictures showed Mr Markle in a series of wedding-related activities - including being measured for a wedding suit - apparently unaware he was being photographed.
He is reported to have said that he did not want to cause his daughter embarrassment by attending the wedding.
Samantha, who has not spoken to her famous sibling in three years - and will not be attending the wedding - told Good Morning Britain the photographs had been done with "good intention" and not for money.
She previously admitted the pictures had been her idea, in order to portray a positive image.
She said that Mr Markle, 73, had "a moral right to set the record straight and defend himself", as the media had "raked him over the coals" and "distorted his image".
"I wanted to see him go (to the wedding). But clearly the priority should be whether or not it's safe for him to do that," she added.
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