Speaking for the first time since Archie was born in May, Mr Markle, 75, told the Mail on Sunday: "Of course I'm disappointed not to see Archie.
"I had hoped becoming a mother would mellow Meghan and she would reach out."
He added: "I would like them to send me a picture of Archie so I can frame it and put it on my wall beside the one of Meghan.
"Isn't that what any grandfather would want? I'd love to know if he's got the famous Markle nose."
Mr Markle said the rift between the father and daughter is a recent occurrence.
"There was never any problem between Meghan and me until recently,” he claimed.
"I've stayed quiet because Meghan was pregnant with Archie but I'm speaking out now because they and their PR people are continuing to ghost me out of her life.”
His comments appear to indicate that the rift between the duchess and her father is far from being resolved.
It’s believed that Meghan hasn’t spoken to her dad since her wedding to Prince Harry last May.
Since the wedding, Mr Markle has given numerous interviews to the press and TV channels, including sharing excerpts of a five-page letter Meghan wrote to him in August last year.
In the most recent interview he claimed the royal couple had failed to enquire after his health following a heart attack.
"All I'd like to ask people is, 'How would you feel if your father had a heart attack?'“ he said.
"I texted them [Meghan and Harry] from my hospital bed and told them where I was. I have the texts to prove it.
"Yet never once has Meghan ever asked me how I'm doing. Not then, not since."
Mr Markle also disputed claims made by Meghan that she had funded her own university tuition.
In her first major speech as a royal, the duchess told students at the University of the South Pacific in Fiji last year that she used financial aid to see her through her studies.
"The journey of higher education is an incredible, impactful and pivotal one,” she said.
"I am also fully aware of the challenges of being able to afford this level of schooling for many people around the world, myself included.
"It was with scholarships, financial aid programmes, and work study from my earnings from a job on campus that went directly towards my tuition that I was able to attend university, and without question it was worth every effort."
But Mr Markle claims to have paid "every penny" of Meghan's $200,000 (£165,000) tuition at Northwestern University in Evanston, Illinois.
"I paid every penny of her tuition and I have the bank statements to prove it,” he told the publication.
In the interview Mr Markle also appeared to reference recent criticism of the royal couple’s recent trips on private jet.
“I’ve still never met my son-in-law,” he said. “Why couldn’t Prince Harry have got on a plane and flown to see me to ask for her hand?
"He obviously has no problem getting on private jets.”
Following the release of the interview, royal sources claimed Meghan has made the “heartbreaking” decision to continue to remain estranged from her father and will likely never let him meet Archie.
Though Meghan’s mother, Doria Ragland, was mentioned in a statement when Archie’s birth was announced, Thomas was not included.
And though he did release a statement congratulating his daughter on the birth of her son, it’s believed he hasn’t had any other contact with her, a situation that now looks set to continue.