The First Minister wrote to Mr Raab earlier this month after meeting the Johal family, urging him to engage with them and provide direct assurances the Foreign Office was raising the allegations of torture and concerns he will not receive a fair trial.
She wrote: “I would ask you to personally meet with the family to hear their concerns.”
However, in his response Mr Raab failed to acknowledge the request, instead outlining how a consular team were in touch with the Johal family and the case was being handled by Lord Ahmad, the minister for south Asia.
Mr Raab said he shared Ms Sturgeon’s concerns abut the delays to legal proceedings and had discussed the case with the Indian minister of external affairs early this month, but failed to reveal the outcome of that conversation.
In his letter, the foreign secretary also admitted that Boris Johnson had been unable to raise the case with India’s Prime Minister Narendra Modi during a virtual conversation in May, which the Johal family had hoped would see Mr Johnson secure Jagtar’s release.
Mr Raab writes the Prime Minister “did secure agreement to a 2030 road map for India-UK future relations, which includes an objective to promote closer co-operation in consular matters and to resolve long running or complex consular cases, of which we would consider Mr Johal one”.
Jagtar Singh Johal, known as Jaggi, has been in prison for four years after being “effectively abducted” in Punjab by plain-clothes police officers while out shopping ahead of his wedding.
His lawyers say he was placed under a hood, taken to an undisclosed location and tortured for several days, before being forced to sign blank sheets of paper.
Indian authorities claim he is detained under anti-terror laws and has been charged with murder and criminal conspiracy.
His brother Gurpreet, who has led the campaign for Jaggi's release, said Mr Raab’s letter “was a lot of blather”.
He said: “In the end it comes down to this. The foreign secretary won't meet me, the Prime Minister didn't raise my brother's case when he had the chance and the government won't seek his release. That they claim to be doing so much for him almost makes it worse.
"There's no urgency. They talk about the Prime Minister 'securing agreement' to a '2030 road map' when he could have been securing my brother's release. Jagtar needs the UK's help now."
The Johals are being supported by Reprieve, the non-governmental organisation that investigates human rights abuses. Its regional death penalty expert Harriet McCulloch said: "For the whole two years that Dominic Raab has been foreign secretary, Jagtar has been wasting away in an Indian prison.
"Can he really not have time to simply meet Jagtar's family, hear their concerns and help them get him home?
"We are talking about a young British man unjustly locked up overseas and facing a death sentence. A decisive intervention from the foreign secretary could save his life.”