This past Sunday, Governor Inslee said, he made his views clear at a governors’ breakfast with Scott Pruitt, the head of Mr. Trump’s Environmental Protection Agency and the president’s point man in undoing climate change rules. “I told him, you’re killing my state,” Governor Inslee said of his conversation with Mr. Pruitt. “My state is going up in smoke because our forests are involved in these catastrophic fires.” He added, “I told him, this administration is not just irresponsible but morally reprehensible.”

An E.P.A. official didn’t immediately respond to a request for comment from Mr. Pruitt.

The global interest in Governor Inslee’s carbon tax push comes as several state leaders are signaling to the world that they intend to act on climate change with or without the Trump administration. More significantly, the state has drawn attention as a major global economy that is willing to buck conventional wisdom by taking on a carbon tax at all.

“This is very significant. Washington is not just any state, it’s home to so many global companies: Starbucks, Microsoft, Amazon,” said Erik Solheim, the head of the United Nations Environment Program, speaking by phone from New Delhi, India. “It is part of the signal to the world that the major states, big business are in disagreement with the U.S. withdrawing from the Paris climate agreement, and they are doing something about it.”

Opponents of climate change policy, including the fossil fuel industry and allies of the Trump administration, also see the potential for a Washington carbon tax as seminal. “If it works in the state of Washington, it’s going to be tried in 10 states next year and 35 states the year after that,” said Michael McKenna, a Republican energy lobbyist and political strategist who advised Mr. Trump’s campaign and transition. “If Inslee’s successful, it will be a game-changer. Everyone will take it and copy it and be off and running.”

Already, carbon tax bills have been introduced in the legislatures of Utah, Maryland, New York, Hawaii, Rhode Island, Vermont, Maine and Washington, D.C.