Glenn Taylor was filmed by a colleague pushing the 170 million-year-old red rock in Goblin Valley State Park and celebrating afterwards.
The two scouts say they have received death threats after the video was posted online.
The two men argue the rock was loose and could have fallen on a passer-by.
Utah State Parks spokesman Eugene Swalberg said the state authorities were considering bringing the charges after the incident.
“This is not behavior that is appreciated or should exist in state parks,” Eugene Swalberg told the Deseret News.
“This has been formed for literally millions of years, and it’s supposed to last for a long time. It doesn’t need individuals doing the work of Mother Nature.”
The Boy Scouts of America – who have millions of members across the country – also condemned the action, warning that it would take “appropriate” measures.
The scout leaders said the stone was pushed over because of safety concerns.
However, scout leader Dave Hall told the Salt Lake Tribune: “I think we made the right decision, but probably the wrong method.
“We take full responsibility for whatever mistake we made, and we’re open to whatever that means from the state, from the Boy Scouts’ office, etc.”
In the video, Glenn Taylor is seen congratulating a colleague after the mushroom-shaped sandstone rock – known as a “goblin” – was toppled.
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