The Duke of Cambridge has defended the monarchy against accusations of racism made by the Duke and Duchess of Sussex, saying: “We’re very much not a racist family.”

Prince William said he had not yet spoken to his brother since Harry and Meghan launched their attack on the family and institution in an interview with Oprah Winfrey broadcast in the US on Sunday.

William’s statement is the first public personal comment by any member of the royal family since the allegations, and gives an insight into the hurt that members of the family are understood to feel over the matter.

The Queen’s aides issued a very brief official statement on her behalf on Tuesday.

William and the Duchess of Cambridge toured School 21 in Stratford, east London, on Thursday to mark the return of children to classes this week and the rollout to secondary schools of a mental health project that Kate launched in primary schools in 2018.

As William left, a reporter asked him: “Is the royal family a racist family, sir?”

The duke, with the duchess by his side, replied: “We’re very much not a racist family.”

Of Harry, he said: “I haven’t spoken to him yet, but I will do.”

Meghan told Winfrey that when she was pregnant there were “concerns and conversations” about how dark their son Archie’s skin might be. The couple attributed the comments to an unnamed member of the royal family, said not to be the Queen or Philip. Harry said he felt “awkward” and “shocked”.

Meghan also suggested that Archie had been denied his birthright of the title of prince because he was mixed race. She described feeling suicidal and said she was denied help for her mental health when she sought it.

The Queen said in her statement she was “saddened” that her grandson and his wife had found life as working royals so challenging. She said the issues raised, “particularly that of race”, were “concerning”. “While some recollections may vary, they are taken very seriously,” the statement added. The matter would be addressed “by the family privately”.

The interview will have done nothing to heal the rift between the brothers. Meghan accused Kate of having made her cry during a row over flower girl dresses before her wedding in May 2018. Kate had apologised and sent flowers, Meghan added.

Prince Harry on his wedding day in 2018, with Prince William, right, by his side.

Meghan said she was disclosing this to correct erroneous reports that it was she who had made Kate cry. “I am not sharing that piece about Kate to be disparaging about her,” she said, adding that Kate was “a good person”.

However, any attack on his wife will surely have angered William. Harry described their relationship as “space” at the moment, so it is not surprising that he and William have not yet spoken since the broadcast.

It is rare for a senior royal to speak of such intensely private matters during a public engagement, as William has done. The brothers’ relationship has been in difficulties since William reportedly advised Harry that he should “slow down” and not rush into things with the ex-Suits actor.

According to Finding Freedom, an unauthorised biography of the Sussexes, Harry was angered by what he perceived as his brother’s “snobbish” attitude towards Meghan.

The ITN journalist Tom Bradby, a friend of the brothers, and who interviewed the Sussexes during their southern Africa tour, has previously written that the fallout began at the time of the wedding. “Really damaging things were said and done,” he wrote.

When asked about William by Winfrey, Harry said: “The relationship is space at the moment.” He added: “And time heals all things, hopefully.”

“I love William to bits, he’s my brother, we’ve been through hell together, we have a shared experience, but we were on different paths.”

The brothers are due to come together to unveil a statue they had commissioned of their mother on 1 July, what would have been her 60th birthday. Amid the fallout from the interview, that prospect now looks far from certain.

The royal family dispute was raised in the House of Commons on Thursday when the Conservative MP Sir David Amess asked if time could be found for a debate on the role of the monarchy.

“During such a debate, I’d very much hope that the argument could be made that it is never wise for a family dispute to be aired in public with everyone getting damaged and hurt by the fallout,” Amess said. He added that any debate could celebrate the Queen’s reign as monarch for 70 years.

The Commons leader, Jacob Rees-Mogg, responded by saying: “Were we to have a debate to praise our sovereign lady, it would take up all the legislative time available in this house.” He then recited the first two verses of the national anthem.