Fayed, who died last year aged 94, is accused of multiple rapes and sexual assaults by several women who worked for him – many of whom felt unable to report what had happened until recently.
Dozens more women have been in touch since the BBC programme aired last week.
Natacha, part of the Justice for Harrods survivors group, said Dr Coxon had questions to answer about the medical examinations that she carried out on behalf of Fayed.
“The examinations carried out by Dr Coxon were intrusive and wholly unnecessary,” Natacha said.
“They also resulted in many employees’, including my own, confidential medical information being inappropriately shared within Harrods. This should not have happened.”
She said the group would expect the regulator to investigate the allegations made in the BBC documentary.
A GMC spokesperson said the allegations relating to the medical staff were “deeply concerning”.
“If we identify any potential fitness to practise concerns about individual doctors, we will thoroughly examine all relevant information and take action as appropriate,” they added.