“Allegations, accusations have been made – they obviously need to be looked at,” said Jonathan Ashworth, the party’s health spokesman.
Asked how the investigation should be carried out, Mr Ashworth replied: “That is a matter for the Palace and they will need to come forward with a process.”
And – after more than 24 hours of silence from the royal family – he told BBC Radio 4: “I think people will expect them to make a response.”
The duchess claimed that an unnamed member of the royal family voiced concerns about the colour of their unborn baby’s skin.
The couple made a string of other claims, including that Meghan’s pleas for help when she was feeling suicidal were ignored by Palace courtiers.
It was reported that the Queen refused to sign off a statement that officials had hoped would lower tensions by stressing the family’s love and concern for the couple, to allow more time to consider her response.
Some have noted that more time has passed than it took for the Palace to respond to last week’s bullying allegations against Meghan – when it announced it would investigate her behaviour.
Buckingham Palace has been thrown into turmoil, particularly by the claim of comments made about Archie’s skin colour – after Ms Winfrey clarified that they were made by neither the Queen nor Prince Philip.
Senior ministers have privately expressed fears that the accusations will eventually be made directly against a very senior member of the royal family, causing huge damage to the monarchy.
Ducking the controversy at a press conference, the prime minister yesterday said only that he had “the highest admiration for the Queen and the unifying role that she plays in our country”.
Robert Buckland, the justice secretary, also refused to add to what he called “the running commentary”, in interviews on Tuesday morning.
“It is not right for me, as a member of the government, to start commenting on matters that are for the royal family,” he said.
In contrast, Keir Starmer said the controversy was “bigger than the royal family”, saying: “For too many years we have been too dismissive and too willing to put these issues to one side.”
One minister has commented, Zac Goldsmith – a close friend of the prime minister – targeting the duke and duchess by tweeting: “Harry is blowing up his family. What Meghan wants, Meghan gets.”
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