After the interview Mrs Bush Hager joked that she should call him "brother-in-law Harry".
She added that she gave her sister Barbara, 34, the founder of Global Health Corps, a health and human rights organisation, the "heads up" about what she was going to do and that her sister was "humiliated".
Elsewhere in the interview the Prince said that his grandmother the Queen is "definitely my boss" and agreed that when she talks, he listens.
He said: "My grandma has always been the boss. But my god, she gives amazing advice and she let's us cruise around doing what we think is right."
The Prince added that mistakes were "part of growing up".
He said: "You've got to find your own path and if you slip off people don't tell you you've slipped off then you sure as hell work it out yourself."
In a separate interview, Michelle Obama said Prince George was the "cutest baby" and recalled seeing him with Prince Harry during a visit to Kensington Palace last month.
She said: "I have to say that the most precious thing, if you haven't fallen in love with it already, is to see him lift his nephew."
She added that she was surprised to hear the toddler say: "Uncle Harry, you're so quiet."
Harry had an on-off seven-year relationship with Davy, a former lawyer who is supposed to have felt unable to take on everything that goes with being the wife of a Royal
Florence Brudenell-Bruce
Tatler cover girl who dated Jenson Button, modelled lingerie and once starred in a Bollywood movie. The pair dated briefly in 2011
Cressida Bonas
23-year-old daughter of four times married Lady Mary-Gaye Georgiana Lorna Curzon, a sixties It girl. A Burberry model, Cressida went to Stowe school and Leeds University. Once described by Tatler as "really pretty" and "really nice"