Helen Holland from Birchanger in Essex, was struck at the junction of West Cromwell Road and Warwick Road in Earl's Court, west London, on 10 May.
Her family said the 81-year-old fought "for her life for nearly two weeks... but irreversible damage to her brain finally ended the battle today".
An investigation into the crash by the police watchdog is ongoing, BBC News reports.
Buckingham Palace previously said Sophie the Duchess of Edinburgh's "heartfelt thoughts and prayers are with the injured lady and her family".
Ms Holland had been in London visiting her sister at the time of the crash.
Following the crash, her family said she was in a coma, and on Friday the police said she remained in a critical condition.
Announcing her death, her son, Martin Holland, said his mother had died after "suffering multiple broken bones and massive internal injuries".
Chief Superintendent Richard Smith, head of the Met Police's Royalty and Specialist Protection unit, said the "tragic outcome is being felt by colleagues across the Met" and "thoughts are very much with the woman's family and loved ones at this sad time."
"Officers know that their actions, both on and off duty, are open to scrutiny and following our referral of the incident, the IOPC launched an independent investigation - we continue to co-operate with and support that enquiry," he said.
The Independent Office for Police Conduct (IOPC), which investigates the most serious incidents involving officers, has previously said that, as part of its investigation into the incident, CCTV from nearby properties and police body camera footage would be examined.
It added that officers who were on the scene would be interviewed and appealed for other witnesses to come forward.
The body also said it had been in touch with Ms Holland's family to "explain our role and will keep them regularly updated as the investigation progresses".
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