The revelation means no-one in the Scottish justice system has seen the full report, prompting concerns that lessons have not been learned since Ross Monaghan stood trial accused of Mr Carroll's murder.
Prosecutors and the Scottish Police Authority's forensic services have only seen the conclusions and recommendations of the report, which was carried out by the Home Office Forensic Science Regulator.
The report was ordered by the police authority after Mr Monaghan walked free in 2012 after a high court judge described the case's forensics as "horrendous".
The Crown is blocking the report's release because the Carroll case is ongoing, with a suspect, Billy Paterson, believed to be on the run abroad.
However, Labour justice spokesman Graeme Pearson warned that the situation could affect public confidence in the justice system.
He said: "I am surprised there is a suggestion that no-one in the Crown Office has read this report in its entirety. The public deserve to know its content and impact.
"As the suspect is currently assumed on the run abroad, we have no way of knowing when the case will be concluded - if ever. We need to know what steps have been taken regarding the regulator's investigation and what was the result.
"As it stands, this case will do nothing to increase public confidence in the openness and transparency of Scotland's justice system."
Scottish Conservative justice spokeswoman Margaret Mitchell said: "In a single police force, without the checks and balances of another force to conduct internal investigations, transparency is vital. That transparency must extend to the Crown, and I'm surprised this hasn't happened."
Mr Monaghan walked free from court amid claims the police had undue influence on the forensic scientist in the case. Judge Lord Brailsford also criticised a police raid on Mr Monaghan's Glasgow home because of possible contamination of evidence.
The court also heard that the DNA of a lab worker was found on the suspected murder weapon, a hand gun, but he had never touched it. Derek Ogg QC, Mr Monaghan's advocate at his trial, said: "What was the point of instructing this report if no-one is going to see it [and] if lessons aren't going to be learned?
He added: "We need to know what was in this report. We need to know exactly what happened."
Contrary to the Crown's view, the lawyer added, not releasing it may do more harm. he asked: "How could this prejudice proceedings? This is what came out in court. It's in the public domain.
"Any future jurors will know there have been doubts cast over the scientists and the police in the case, so how does not publishing the report do any good? Surely releasing it will offer reassurance the issues ... are being addressed?"
Brian McConnachie QC, chairman of the Faculty of Advocates Criminal Bar Association, said: "It's particularly important that the prosecution authorities are aware of what's contained in the report. If a report details some mistakes that were made in the past, it seems to be common sense that the Crown would want to know what these mistakes were.
"It seems not only has the Crown blocked its release to the public, even more than that, they appear to have blocked its release to the very people who instructed the report."
A Crown Office spokesman said: "There is a significant risk to ongoing live proceedings if the report is released. It would not be appropriate to speculate on matters connected to an ongoing case or comment further at this time."