Operator CHC suspended flights of its fleet of Super Puma AS332 L2 helicopters at the weekend after one of the aircraft plunged into the North Sea off Shetland killing four people.
CHC announced the suspension to "take stock of any implications associated with Friday's accident."
RMT general secretary Bob Crow demanded: "The entire Super Puma fleet must remain grounded until the causes of this latest event are established and dealt with thoroughly to the union's satisfaction."
He warned that offshore union RMT "will support any member who refuses to board any suspect aircraft type in light of this disaster."
Workforce confidence in the Super Puma type aircraft was "severely dented" after two ditching events last year and a fatal accident in 2009, he added.
Since then both RMT and Unite have worked with all sectors of the industry to address the concerns of its members and rebuild that confidence.
But he added that the crash had "undone all of that work" and he anticipated "an outpouring of anger."
RMT is demanding the lifting of the ban on union access to the offshore workforce, which Mr Crow described "an infringement of basic rights" that "makes a mockery of pledges on offshore worker safety."
Unite Scottish secretary Pat Rafferty said that this latest disaster was the fifth such incident to occur in the last four years and the second to result in fatalities. "It's unacceptable and it can't go on," he said.
CHC also suspended all British commercial flights of the three other models in the Super Puma range after the offshore industry's Helicopter Safety Steering Group urged the precautionary measure until there is "sufficient factual information" to resume flights.
The Super Puma which crashed on Friday evening two miles east of Sumburgh airport was flying from the Borgstein Dolphin platform with 16 workers for oil company Total and two crew aboard.
Duncan Munro, 46, from Bishop Auckland; George Allison, 57, from Winchester; Sarah Darnley, 45, from Elgin; and 59-year-old Gary McCrossan, from Inverness, were the four workers killed.