The Rail Maritime and Transport union said there is increasing public and political support for an inquiry, covering onshore as well as the North Sea offshore industry.
The union said the issue has become more urgent following the helicopter crash in Glasgow last month, in which 10 people died.
RMT general secretary Bob Crow said: "In light of recent events, including the helicopter tragedies both offshore and onshore, and the grounding of helicopters in many areas of the country, the case for a full public inquiry dealing with helicopter safety, and the associated safety issues in the North Sea field, is overwhelming.
"Cost should not be a barrier to establishing the full facts and assisting us in developing a safety regime that takes us forward in an atmosphere of trust and co-operation."
The union has written to First Minister Alex Salmond, seeking his full support for a public inquiry.
There have already been calls for all passenger-carrying helicopters to be fitted with black box recording devices in the wake of the Clutha crash on November 29. Three people inside the aircraft and seven inside the Clutha pub died when the Eurocopter EC135 crashed into the roof of the building.
Air crash investigators are still trying to establish the cause of the crash but an initial report found no evidence of "major mechanical disruption".