Three out of four trains are now either late or cancelled.
Ross Greer MSP quizzed Ms Sturgeon during First Minister’s Questions last week.
He said: “The latest ScotRail performance statistics show that on the Milngavie line, three in four trains are late or cancelled.
“Does the First Minister believe that ScotRail are fulfilling their obligations to passengers on a line where they can’t get three out of every four trains reliably?”
The First Minister responded: “No, I don’t think that’s acceptable for passengers.
“That performance level is not acceptable, which is why we are working with ScotRail and it’s why we set stringent expectations for ScotRail to take the action that improves performance and improves the passenger experience.”
Mr Greer has been at the forefront of a community campaign to reinstate the second track on the line, which he and other local politicians have said would help prevent delays and improve services as far as Edinburgh.
He said: “These services are affected daily by delays originating between Milngavie and Westerton.”
East Dunbartonshire Council recently agreed, with all-party support, to ask consultants to build the case for redualling.
This followed an expert report commissioned by Mr Greer, which laid out the improvements this would bring, and also included the possibility of building the long called for Allander rail station.
Speaking after First Minister’s Questions, Mr Greer said: “Three out of every four trains are delayed or cancelled on the Milngavie line.
“It has been the worst performing in Scotland for far too long.
“Passengers know it’s unacceptable, Network Rail’s own report acknowledges it’s unacceptable and now we know the First Minister agrees.
“Yet the Scottish Government wouldn’t even part-fund the feasibility study into redualling from the Local Rail Development Fund secured by the Greens for projects precisely like this one.
“The evidence is overwhelming that redualling will cut delays and cancellations and allow an Allander Station to be built.
“We know that the work could be done relatively easily as space is still there from when the second line was removed a generation ago.
“I’ll continue to push for action, including at a forthcoming meeting with the Transport Secretary.
“I encourage as many people as possible to sign our petition at https://greens.scot/milngavieredual.”
Local Conservative MSP Maurice Golden and Conservative joint council leader for East Dunbartonshire Andrew Polson, also called this week for urgent action to be taken over the Milngavie line.
Councillor Polson said: “Milngavie station is vital to the local economy, so it is absolutely imperative that the SNP Government outline how they plan to urgently improve performance at the station.
“We are constantly told services will improve, but the proof is in the statistics which show that almost three quarters of trains don’t arrive on time.
“That is completely unacceptable, which makes the dismissal of redualling the line all the more disappointing.
“I will continue to push the SNP Government to secure improvement for services at Milngavie.”