The 74-year-old was caught up in controversy days before the wedding of Harry and Meghan after he allegedly staged paparazzi photographs of himself.
In a picture of the card published by the Sunday Mirror, Samantha Markle wrote: “Holidays are a time for family and are sentimental.
“Dad has been trying to contact you and is very hurt because you are avoiding him.
“Life is short and you know Dad is wonderful. Please give him joy in his last years by showing you love him. The time is now.”
Speaking from San Diego during an interview with Good Morning Britain earlier this week, Mr Markle made a public appeal to his daughter to get in touch.
He said: “I’m hopeful that soon something will be resolved and we’ll be talking. This can’t continue forever.
“People are saying I should be quiet about this and not speak, if I don’t speak, they will speak to me. That hasn’t happened. I don’t plan to be silent for the rest of my life.
“I love my daughter very much and she has to know that. I would really appreciate if she would just call me, reach out to me somehow, send me a text, just say you’re there and you’re hearing me.”