She also suggested Archie was not made a prince because of his race, even though rules set by George V meant he was not entitled to be one.
Oprah asked: “Do you think it’s because of his race? I know that’s a loaded question.”
Meghan said: “In those months when I was pregnant, all around this same time, so we (had) the conversation of he won’t be given security, he’s not going to be given a title.
“And also concerns and conversations about how dark his skin might be when he’s born.”
Buckingham Palace said they were “saddened” by Harry and Meghan’s claims, but added that “some recollections may vary”.
Meghan also hinted there could have been a missed opportunity for her biracial links to help change the Royal Family’s image in the Commonwealth during the Oprah interview.
She said: “The Commonwealth is a huge part of the monarchy, and I lived in Canada, which is a Commonwealth country, for seven years.
“But it wasn’t until Harry and I were together that we started to travel through the Commonwealth, I would say 60 percent, 70 percent of which is people of colour, right?”
She said: “I’d like to be a queen of people’s hearts, in people’s hearts, but I don’t see myself being Queen of this country.”
She also spoke openly about her own mental wellbeing, including suffering from post-natal depression after the birth of Prince William, and the lack of understanding she had received from members of the Firm.
Diana said: “Well, it gave everybody a wonderful new label – Diana’s unstable and Diana’s mentally unbalanced. And unfortunately, that seems to have stuck on and off over the years.”
Other topics included dealing with the “daunting” media attention and being followed by photographers, as well the books written about her family.
Over the past few months, Harry has similarly spoken publicly about his own struggles with media attention and his mental health.
He also told Oprah his family did not speak about Diana’s death in 1997 and expected him to just deal with the resulting press attention and mental distress.
He said: “My father used to say to me when I was younger, he used to say to both William and I, ‘Well it was like that for me so it’s going to be like that for you’.
“That doesn’t make sense. Just because you suffered doesn’t mean that your kids have to suffer, in fact quite the opposite – if you suffered, do everything you can to make sure that whatever negative experiences you had, that you can make it right for your kids.”