He said the scenes at the vigil were “completely unacceptable”, despite having received assurances from Scotland Yard last week that it would be policed “sensitively”.
“In my view, this was not the case,” he said.
Mr Khan said: “I asked the Commissioner and Deputy Commissioner to come into City Hall today to give me an explanation of yesterday’s events and the days leading up to them.
“I am not satisfied with the explanation they have provided.
“I will now be asking Her Majesty’s Inspectorate of Constabulary (HMIC) to conduct a full independent investigation of events yesterday evening and in previous days.
“I am also asking the Independent Office for Police Conduct (IOPC) to investigate the actions of police officers yesterday evening.
“It is vital that these events are not allowed to undermine the powerful calls since Sarah’s murder for meaningful action to finally stop men inflicting violence on women.
“It was clear before yesterday that there isn’t adequate trust and confidence from women and girls in the police and criminal justice system more widely.
“Further steps must now be taken to address this.”