It came after an inquiry into the deaths of the 14 oil workers and two crew in the Bond Super Puma crash on April 1, 2009 concluded the tragedy might have been prevented.
Relatives, who have criticised their five-year wait for the fatal accident inquiry (FAI), have called for a criminal inquiry.
But the Crown Office defended its decision not to prosecute, saying evidence put to the six-week FAI in Aberdeen had not changed its view that there was insufficient evidence.
Sheriff Principal Derek Pyle, who heard the FAI, said the precautions which might have avoided the deaths included following agreed procedures and communications.
He found that Bond Offshore Helicopters had failed to remove and examine a part of the gearbox just a week before the crash - a task specified in the aircraft maintenance manual - because of a failure in communication with the manufacturer, Eurocopter. Had the company done so, he said, the accident may have been avoided.
The Sheriff Principal said the cause was a catastrophic failure of the helicopter's main rotor gearbox, caused by spalling (fracturing of metal) in the gearbox.
"The essential fact is that everyone in the company well knew that maintenance must be done by the book. On one occasion, that fundamental rule was broken. It resulted in the failure to detect a significant fault in the helicopter's gearbox, which possibly, but only possibly, resulted in the crash," he said.
Caroline Doyle, 35, whose father Raymond Doyle, 57, of Cumbernauld, North Lanarkshire, died in the crash, said: "How this is not deemed a breach of health and safety is beyond comprehension.
"My dad went to work, and he didn't come back because, as far as I am concerned, someone didn't take a gearbox and check. I don't want to talk about compensation; all we want is the evidence heard, properly and justice done for our loved ones. We want someone to be held accountable."
Solicitor advocate Tom Marshall, who represented relatives at the FAI, said: "There were numerous opportunities for Bond to have prevented this tragedy. Had they followed the correct procedure the fault in the gear box would have been properly dealt with.
"It's an appalling state of affairs where 16 men can lose their lives while simply returning from work, and yet no-one has yet been prosecuted."
The families have joined trade unions in calling for a full public inquiry into offshore safety in the North Sea. The Crown Office said it would meet with relatives but that the decision not to prosecute was "correct".
A statement said: "For a criminal prosecution to have taken place, the Crown would have to prove its case beyond reasonable doubt. The Sheriff Principal makes clear that a reasonable doubt remained over the technical cause of the crash. The evidence presented during the FAI has not altered the insufficiency of evidence therefore the decision not to hold criminal proceedings remains the correct one."
In a statement Bond Offshore said: "We have always accepted we made mistakes through honest confusion over telephone calls and emails.
"Lessons needed to be learned, lessons have been learned and lessons continue to be learned. We are absolutely committed to continuing to drive safety improvements across the business, and will study the Sheriff Principal's recommendations carefully, along with our industry colleagues."
The victims included captain and co-pilot Paul Burnham, 31, from Methlick in Aberdeenshire, and Richard Menzies, 24, from Droitwich Spa in Worcestershire.
l A Bond-operated Sikorsky S-92 landed safely yesterday at Aberdeen heliport after a technical warning on one of its engines. Emergency services attended as a precaution and checks are now being carried out on the helicopter.