The relationship broke down after controversial staged paparazzi pictures and a series of public outbursts about the Royal Family by the former Hollywood lighting director.
Mr Markle, 74, missed the Windsor Castle wedding after undergoing heart surgery and admitted: “I don’t expect to see her or hear back from her.”
But Meghan’s brother Thomas Markle Jr claims he has written to the Queen calling on her to intervene and help repair the relationship.
He wrote: “I urge you to convince Meg to bring her father over to the Palace for him to meet the Royal Family and spend time with him.
“I recently went to see him after hearing of his heart surgery and his health is not good.
“I fear if Meghan waits too long it will be too late as Tom Sr feels like he won’t see her again.”
He added: “My father deserves to be treated like royalty, like he treated Meghan her whole life.”
Mr Markle Jr, 52, told a US magazine that he fears “Meg may feel like she’s being held prisoner and turning into a zombie”.
Meghan’s father says he has no way of leaving messages for his daughter – two private numbers given by Kensington Palace are no longer in service.
Thomas Jr recently said he wished the Royal Family had embraced the Markles in the same way as they welcomed the Duchess of Cambridge’s family, the Middletons.
He said: “It’s a shame where it has ended up, especially between her [Meghan] and my father.
“She’s capable of doing anything, she has just got to want to do it. If she wants to make amends, by all means, she can do that. But she has got to want to do it.”
Meghan’s father has spoken of his pain at not being able to make a speech at the wedding, saying: “I was going to recount the funny story about how she broke the news to me about dating Harry… ‘Daddy, I’ve met a man. Daddy, he’s English. Daddy, he’s a prince’.
“It was a cute story. Like any father I wanted to say a few words. I was going to thank the Royal Family for welcoming my beautiful daughter into their family. But it was a speech that I was never going to be allowed to give. That hurt.”
Mr Markle said he had hoped Meghan would send a formal gold-embossed invitation.
He said: “I asked Meghan to send invitations to all my family. I told her they wouldn’t be able to come because they would never be able to afford the flight, but that it would be a nice gesture on her part. Of course that didn’t happen.”