The Dunn family are taking legal action against Mr Raab after the 19-year-old died in a hit-and-run crash outside RAF Croughton, Northamptonshire, in August.
They accuse Mr Raab of abusing his power and “robbing their entitlement to justice” by granting diplomatic immunity to Anne Sacoolas, 42, who is believed to have hit the teenager while driving into oncoming traffic.
Ms Sacoolas, whose husband is thought to be a US intelligence officer, has been flown home to the US.
The Foreign Office are challenging the legal action and have invited the Dunn family to drop proceedings.
Mr Raab told Sky News on Wednesday morning that the government will seek legal costs if the Dunns' case over Ms Sacoolas' status goes to court, even though it “pains” him.
"The representative of the family is suing the Foreign Office and they've talked about a litigation strategy in the US as well,” he said.
"They're crowdfunding for that purpose and we just cannot responsibly allow ourselves to be sued without taking the normal action in defending ourselves when the position that the representative and the family are pursuing in law is wrong.
"It pains me because I want to give them the solace of justice in this case. But we also need to protect the taxpayers' money and the legal position that we set out, which is the correct one. So it's very difficult, but that's the position that we're in."
But following the interview, Dunn family spokesman Radd Seiger said they were “livid”, adding: “They are British citizens who have had their entitlement to justice robbed from them by the US Government under the watch of Mr Raab and the FCO.
“Mr Raab knows perfectly well that Mrs Sacoolas never had diplomatic immunity. He has lied to the parents before and he is now lying to the nation.
“[...] If he is so concerned about taxpayers’ money in the litigation then he would come and talk to us to find a resolution rather than risking having the taxpayers themselves having to pay a very expensive legal bill if the FCO lose.
“[...] The parents are concerned that the US and U.K. governments are conspiring and colluding to pervert the course of justice by interfering in the CPS’ charging decision and they will not stand idly by whilst that is happening.”
A GoFundMe crowdfunding page launched by the Dunn family at the start of October has already raised more than £90,000 of its £100,000 target.