Ms Sturgeon said the incident which claimed the lives of Lamara Bell and John Yuill had been the “lowest point” of her time as first minister.
But Ms Bell’s brother, Martin, said it was too soon for the first minister to comment on the case given that an investigation into the circumstances is continuing.
And he questioned the SNP leader for giving her backing to former Chief Constable Sir Stephen House over the incident.
Mr Yuill, 28, and Ms Bell, 25, died after their car crashed on the M9 in July last year.
An initial call from a member of the public was not properly logged by the police, who did not find the vehicle until three days later.
Ms Bell, a mother of two, was still alive at that time, but died later in hospital.
An investigation into the incident is currently being carried out by the Police Investigations and Review Commissioner (Pirc).
In comments published today, Ms Sturgeon said the “dreadful failure” that led to the deaths had been “tragic beyond words”.
But she defended the SNP’s record on policing, including the decision to merge the country’s eight regional forces in 2013 to form Police Scotland.
However, Mr Bell said the police’s delay in responding to the crash would not have happened had cuts not been made to control rooms under the national police force.
He said: “The question my family would like put across to Nicola Sturgeon is does she regret backing these cuts and Sir Stephen House?
“Yes, we agree that not everything will go right but do people deserve to die while they put in place all these cost-cutting changes to a service that worked fine before all this change?”
Mr Bell said that before Police Scotland the initial call about Mr Yuill’s car being off the road would have been handled better and passed to local officers to investigate.
He added: “If this had happened that day it could have been so different. How many more people have to die needlessly before it all gets put right? Respect and trust have gone. I have people I don’t know mail me everyday telling me about bad experiences they have had with (the non-emergency number) 101.
“We are nine months on and still have no answers. We have been left hanging, going over the same thing in our heads over and over. My family is slowly starting to crumble with each day that passes.”
Mr Bell said the first minister had been premature to start talking about the “positives” when “the problem is still there and this could happen again at any time”.
The Pirc is expected to conclude its investigation into the incident in the coming weeks. A report will be submitted to the Crown Office.