Prince Harry and Meghan told Oprah that before their son was born, an unnamed member of the Royal Family voiced concerns about his skin colour.
Meanwhile, US President Joe Biden’s spokeswoman praised the courage of anyone speaking about mental health.
White House press secretary Jen Psaki described Harry, 36, and Meghan, 39, as “private citizens” who were “sharing their own story in their own struggles”.
BBC royal correspondent Jonny Dymond said the palace’s apparent strategy of silence “faces a challenge”, adding that the institution is always “relatively miserly” in its comments on such issues.
“It generally clams up and says ‘that’s none of your business’,” he added.
In their interview, the couple spoke about a range of topics, including racism, mental health, and their relationships with the media and other members of the Royal Family.
The duchess said that she found royal life so difficult that at times she “didn’t want to be alive any more”, and when she approached the institution for help, she did not get it.
Meghan – the first mixed-race member of the modern Royal Family – added that a low point was when Harry was asked by an unnamed royal family member “how dark” their son Archie’s skin might be
Labour’s shadow education secretary Kate Green said Meghan’s claims were “really distressing, shocking”.
She told Sky News: “If there are allegations of racism, I would expect them to be treated by the palace with the utmost seriousness and fully investigated.”
Labour leader Sir Keir Starmer said that Meghan’s allegations about racism and a lack of mental health support should be taken “very seriously”.
Prime Minister Boris Johnson declined to comment on Monday, but said he has “always had the highest admiration for the Queen and the unifying role she plays”.
He said “when it comes to matters to do with the Royal Family, the right thing for prime ministers to say is nothing”, after being asked specifically if he believed the Royal Family was racist.
The two-hour interview was broadcast on Sunday by CBS in the US and was screened in the UK on ITV on Monday night.