Mr. Biden recently predicted a “thaw” in U.S.-China relations, but last week he called Mr. Xi a dictator and then stood by it, rankling China. When Secretary of State Antony Blinken visited Beijing this month to lower the temperature, Republicans blasted him. Mr. Biden’s G.O.P. challengers are already calling him soft on China ahead of the 2024 election. “The public climate places a ceiling on where the anticipated thaw could lead,” said Jessica Chen Weiss, a Cornell political scientist.

Public opinion may already be pinching Mr. Biden’s strategy. While advising the State Department from 2021 to 2022, Ms. Weiss advocated a “framework for peaceful coexistence” — deterring China more than provoking it. But, she said, senior administration officials were skeptical that Americans would support anything less than “responsibly managing the competition,” a catchphrase officials use to describe its current approach. “That’s an example of, I think, the indirect influence that the public climate — the discourse, not just the polls — has,” she said. (The White House did not comment on her appraisal.)

Chinese public opinion — which has become similarly negative and hawkish toward the U.S. under Mr. Xi — may also impede de-escalation. Academic research suggests that public opinion can drive leaders’ decision-making even in countries where politicians aren’t democratically elected. “There’s this public outcry for leaders to do something,” Mr. Kertzer said. “And then you end up in a situation where escalation on one side leads to escalation on the other.”

‘We’re already there’

Does that mean the U.S. and China are destined to grapple, Cold War-style, for decades? Not necessarily. Still, frosty relations could become self-fulfilling. A Cold War mentality in both countries could make escalation over Taiwan more likely. “Public opinion data right now suggests if China were to invade Taiwan, there would be strong responses in the U.S.,” Mr. Kertzer said. It could also hurt U.S. allies and businesses that rely on China’s economy, and could close down cooperation and diplomacy. And anti-China sentiment appears to have fueled a rise in attacks against Asian Americans.

Others think a Cold War framework can help keep tensions from turning hot. “We are already involved with China in a worldwide competition,” Mr. Daly said. “I am not advocating or predicting a cold war. I am saying descriptively that we’re already there.” Admitting as much, he added, “can inspire peaceniks as much as it inspires the hawks.”

But if diplomatic friction and mutual suspicion persist, debates over terminology could become beside the point. “The conception at the macro level is that we are really in a serious competition,” Mr. Herrmann said. “Now the public has followed. And it’s not like you can turn this ship around overnight.”