The victim of the incident has been named as Helen Holland, 81, from Birchanger, Essex.
The Met Police said the motorbike collided with a pedestrian at around 3.21pm on Wednesday at the junction of Cromwell Road and Warwick Road in west London, leaving her in a critical condition.
She had been in London visiting her older sister on Wednesday and her family have said she’s now in a coma.
Her son and daughter-in-law Martin and Lisa-Marie Holland told the BBC they were “shocked and sickened at her extensive injuries”.
“She is being well cared for by the NHS who we must thank deeply for their help in keeping her alive,” they added.
BBC News At Ten newsreader Clive Myrie said the family is “calling for a thorough investigation”.
On Thursday, Buckingham Palace said Sophie is “grateful for the swift response by the emergency services”, adding that she will “keep abreast of developments”.
In a statement, the palace said: “The duchess’s heartfelt thoughts and prayers are with the injured lady and her family.
“She is grateful for the swift response by the emergency services and will keep abreast of developments.
“Further comment at this time would not be appropriate while the incident is being investigated.”
The Metropolitan Police has said the pedestrian was taken to hospital following the crash, where she remains in a critical condition.
The Met Police’s Special Escort Group provide armed escorts for Royal Family members, VIPs, protected members of the Government, visiting royals, heads of state and other visiting dignitaries.
It also provides armed vehicle escorts for valuable, hazardous and protected loads – and can include high-risk prisoners.
On Wednesday evening, police said no other injuries were reported and inquiries into the crash were ongoing.
The force said the Directorate of Professional Standards, which investigates officers’ conduct, has been informed as is routine.
Duchess Sophie was also spotted on Friday for the first time since the incident – putting on a brave face as she watched The Riding for Disabled category on Day two of The Royal Windsor Horse Show.
Wearing a beige Barbour coat over a flowery dress and wellington boots she took cover under a cameo umbrella as the jockeys took off.