Dominic Raab has been criticised for his "weak response" to a plea from the first minister to help a Scot detained in India.
The foreign secretary said he was doing "all he can" to help Jagtar Singh Johal, who has been held under anti-terror laws for four years.
He said his ministers were in regular contact with Indian authorities.
In a letter, Ms Sturgeon asked him to meet Mr Johal's family, but he declined to say whether he would.
Mr Johal's brother, Gurpreet Singh Johal, said: "I shouldn't be surprised by now by this sort of weak response, but part of me did hope that the UK government would listen to the first minister and take my brother's case more seriously.
"Nicola Sturgeon made time to meet me; why can't Dominic Raab?"
Mr Johal, from Dumbarton, is accused of conspiring to murder a number of right-wing Hindu leaders.
He was arrested by police in India two weeks after his wedding there in 2017.
He claims he was "falsely implicated" and tortured into making a confession.
After meeting Mr Johal's family earlier this month, Scotland's first minister wrote to the UK foreign secretary telling him the Scot's family was disappointed with the support from the Foreign Office.
She asked him to meet the Johals and said she was concerned about Mr Johal's detention without trial.
She also said she was "deeply concerned" about allegations of torture and mistreatment of Mr Johal while in custody.
Mr Raab replied, saying he "shared her concerns" about delays in legal proceedings.
He said: "We take all allegations of human rights violations extremely seriously and have consistently raised these directly with the government of India, emphasising the importance of a fair trial and the need for Mr Johal's torture allegations to be fully investigated."
'Closer co-operation in consular matters'
He went on to say that the case was being led by Lord Ahmad, minister of state for South Asia and that Lord Ahmad had spoken to the Indian foreign secretary on 23 July.
Mr Raab also mentioned that the prime minister was unable to visit India as planned in May 2021 but that Mr Johnson did "secure agreement to a 2030 roadmap for India-UK future relations" which included an objective "to promote closer co-operation in consular matters and to resolve long-running or complex consular cases" of which he would consider Mr Johal one.
He said the family should liaise with Lord Ahmad on the case.
Human rights group Reprieve has been acting for Mr Johal since last year.
Its deputy director, Harriet McCulloch told the BBC: "The UK government is saying 'we're doing all we can' and that simply isn't true. It's what they told the 140 MPs who wrote to them earlier this year and what they tell Jagtar's family, but to read it in a response to the first minister is still incredibly disappointing.
"This is a very clear case of a British national arbitrarily detained overseas, and it is government policy to seek his release. So what is holding them back?"
The Indian authorities strongly deny the allegations and have said "there is no evidence of mistreatment or torture as alleged".
Mr Johal travelled to India from Scotland in October 2017 for his wedding.
A fortnight later, while on a shopping trip with his new bride in the North Indian state of Punjab, Mr Johal was taken away by police and has been in detention ever since.
His brother Gurpreet said Mr Johal was a peaceful activist and that he is convinced he was arrested because he had written about historical human rights violations against Sikhs in India.
Charge sheets from the Indian authorities outline the case against Mr Johal and a group of men whom they believe were involved in a "series of killings".
It is claimed Mr Johal was a member of Khalistan Liberation Front (KLF), described in the documents as an international "terrorist gang".
He is accused of paying £3,000 to the former head of the KLF to help fund the crimes. The documents claim he "actively participated and had complete knowledge of the conspiracy".
"There are very serious charges against him including murder and abetment of terrorism," an Indian government official told the BBC.
"The seriousness of charges against him have been shared with the British authorities," they added.