The family of Harry Dunn are "absolutely distraught" over comments made by the prime minister, said family spokesman Radd Seiger.
Mr Dunn, 19, died after being hit by a car allegedly driven by Anne Sacoolas, who left the country for the US claiming diplomatic immunity.
Boris Johnson told the BBC that he believed the chances of Mrs Sacoolas being extradited were "very low".
The comments came on Harry's mother's birthday and left her "bitterly upset".
Mr Seiger told the BBC: "The Dunn family are absolutely on their knees and I am still trying to pick my jaw up off the floor.
"It's an outrageous set of comments to make from the leader of this country, whose job it is to represent the people."
Charlotte Charles, Harry's mother, was "bitterly upset and confused" and "absolutely beside herself, on her birthday," said Mr Seiger.
He said the family had agreed with government officials not to comment on the extradition process while proceedings were ongoing.
"I'm hoping Mr Johnson will reflect on the comments he made, they were unhelpful to say the least, " he said.
The family are also taking legal action against the Foreign and Commonwealth Office (FCO).
The claim against the FCO issued on behalf of Mr Dunn's parents - Ms Charles and Tim Dunn - alleged the granting of diplomatic immunity to Mrs Sacoolas was "wrong in law".
New documents, seen by the PA news agency, suggest the FCO will say they did not claim Mrs Sacoolas had immunity.
Northamptonshire Police have now said the force will take part in the family's claim and they will not seek to retrieve any costs.
The US State Department has previously said the extradition request for Mrs Sacoolas is highly inappropriate and would be an abuse.
Can Anne Sacoolas be extradited?
The extradition request is sent via the British Embassy to the US State Department.
A lawyer will then decide whether it falls under the dual-criminality treaty, where the alleged offence is a crime in both countries and carries a prison sentence of at least a year.
The maximum sentence for causing death by dangerous driving is 14 years' imprisonment, although this is usually reserved for the most serious cases.
The US may reject the request for extradition, arguing that Mrs Sacoolas is still entitled to diplomatic immunity.
The crash happened outside RAF Croughton, where Mrs Sacoolas' husband Jonathan worked as an intelligence officer. Mr Dunn died after his motorbike was in collision with a car owned by Mrs Sacoolas.
The 42-year-old left the UK shortly after the crash on 27 August and returned to the US, prompting a justice campaign by the teenager's parents.
Mrs Sacoolas was charged in December by the Crown Prosecution Service with causing death by dangerous driving and the Home Office submitted its extradition request to the US Department of Justice.