The claim is one of the latest of a series of revelations to emerge from the duke's memoir, Spare.
Harry also reveals he only found out Queen Elizabeth had died from BBC News.
The book was scheduled to be released next week, but mistakenly hit the shelves in Spain on Thursday.
According to multiple reports, the instruction from the future king came during a phone call between the pair.
"Then my father called again", Harry wrote. "He told me I was welcome at Balmoral, but… without her [Meghan]."
Charles then apparently went on to explain why Harry shouldn't not bring Meghan with him, but Harry found his reasons "disrespectful".
"He started to explain his reasons, but they didn't make any sense at all, and it was disrespectful as well. I did not tolerate it from him [...] don't even think about talking about my wife like that", Harry is quoted as saying.
His father, Harry claims, was then "repentant".
"[Charles] said, stammering, that he simply didn't want the place to be full of people. Nobody's wife was going to go, not even Kate, he told me, so Meg shouldn't either".
Harry is also said to describe finding out about his grandmother's death from BBC News as his plane was landing in Scotland, after spending his flight thinking about the last times they had spent together.
His aunt, Princess Anne, greeted him upon his arrival at the Scottish estate, and then he saw the Queen's body and spent time considering his loss.
"I stayed in one place without moving, gazing at her for a long time, gathered strength and continued going forward".
Harry also divulged that despite arriving after she had died, he took some time to say things to his grandmother anyway, admitting that he "whispered to her that I hoped she was happy and that she was with grandpa".
Harry and Meghan have always spoken highly of both the late Prince Philip and Queen Elizabeth, the last monarch brought Meghan to Cheshire for some of her first royal engagements after her May 2018 wedding to Prince Harry.
Spare is officially released on 10 January, and ahead of this Harry is set to undertake a series of high profile interviews promoting his new memoir.
In the US, he is confirmed to sit down with 60 Minutes, Good Morning America and The Late Show. In the UK, Harry has spoken to ITV's Tom Bradby, a journalist and broadcaster he has been known to get along well with over the years and even attended his wedding as a guest.