It's the latest flash point in a high-profile ideological battle over fossil fuel emissions pitting the Disrupt Burrup Hub campaign against the gas company and the WA government.
The activist group issued a concerns notice letter to Roger Cook over his comments to media last Friday after the protest at the Woodside annual general meeting.
During the meeting, Emma Heyink and Tom Power, both 17, stood up and called out Woodside chairman Richard Goyder and chief executive Meg O'Neil's children's names as they asked about their futures amid climate change.
"By the time (children's name) and I are 70 (Woodside's) Burrup Hub will still be producing over six billion tonnes of carbon into our atmosphere," Emma said.
West Australian Premier Roger Cook is refusing to apologise to two teenage protesters. (Richard Wainwright/AAP PHOTOS)
Mr Cook was scathing of the pair's behaviour and told reporters the action was intimidating and threatening and should be taken seriously by police.
Lawyers for the teens allege Mr Cook's comments defamed them and were indefensible and damaging.
"This has caused them considerable hurt, distress and anxiety," the letter seen by AAP said.
It also threatened legal action and noted the damages bill could exceed $459,000.
The premier was asked to publicly apologise to the duo and undertake to refrain from further comments and pay $1 for their legal costs by Tuesday.
Mr Cook declined to do so.
"I'll always support people's right to peaceful protest, but it's never okay to bring somebody's kids into it - regardless of the cause," he said in a statement.
"I'm not going to spend any more time responding to this."
It's the latest salvo in a long-running battle over Woodside's multi-billion Burrup Hub gas project in WA's northwest.
The activists last June released a foul-smelling substance to simulate a gas leak at the company's Perth headquarters, forcing about 2000 staff to be evacuated.
They also spray-painted a Woodside Energy logo onto Fredrick McCubbin's painting Down On His Luck at the Art Gallery of WA, and on the front doors of WA parliament building.
Mr Cook was critical of the group for attempting to target Ms O'Neil's family home with paint last August during a protest action that was stopped by counter-terrorism police.
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