He said Meghan had always been “a very controlling person” which was part of her character, but “she’s never been rude”.
Speaking from San Diego during an interview with Good Morning Britain, 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.”
Mr Markle, who wore a jacket and shirt, appeared to show no ill effects from the reported heart attack he suffered in the days before Harry and Meghan married in May.
He was due to meet the Queen in the run-up to the wedding before health issues forced him to cancel.
Mr Markle said he would like Harry’s grandmother to intervene.
“I would appreciate anything she could do,” he said.
Mr Markle, who Good Morning Britain said was not paid for the interview, added: “I would think she would want to resolve family problems. All families, royal or otherwise, are the same and they should all be together – certainly around the holidays.”
Meghan and Kate have been the subject of a series of media reports claiming there is a rift between the two duchesses.
There have been allegations of a dispute over who reprimands staff, tensions during Princess Charlotte’s bridesmaid dress fitting and claims Harry and Meghan’s move from London to Berkshire reflects the widening gulf between the women.
Commenting on negative stories suggesting his daughter was rude to staff, Mr Markle said: “I don’t really believe them. I think they’re gossip and I don’t really believe them. Meghan’s always been a very polite girl. She’s never been rude to anyone to my knowledge.”
He went on to say: “She’s always been a very controlling person and that’s part of her nature. She’s never been rude. The ghosting I don’t completely understand. To my knowledge she’s always been in charge – that’s her nature – but she’s always been polite with it.”
It is thought Mr Markle last spoke to Meghan during the wedding period, and since then his daughter and her husband Harry have announced they are expecting their first child in the spring.
In the run-up to the wedding, Mr Markle was caught up in controversy after he allegedly staged paparazzi photographs of himself and then began commenting regularly to entertainment website TMZ about his contact with his daughter.
The 74-year-old suggested the duke and duchess may have decided to “ghost” him – where a loved one disappears without explanation – after being influenced by press reports about him.
He said: “I’ve only done five interviews, and out of those interviews hundreds of other people come and they get a story and they give a story and they say I’ve said these things.
“I haven’t said so many things, but they’re believing it, and apparently that’s why I’m being shunned or ghosted.”
The 74-year-old also said: “I’ve been quiet for six months and no-one’s returned anything back to me. No-one’s said a word to me. I’m confused by it.”
With Meghan and Harry’s baby due early next year, Mr Markle said: “I think she’ll make a great mum and maybe things will soften a little bit and we’ll connect again.
“There has to be a place for me. I’m her father and I will be the grandfather to the children, so all I can say is that I’m here, she knows it.”
Watch Mr Markle's video below: