A day after the Islamic State released a video showing the beheading of another American, President Obama vowed to destroy the organization, and Vice President Joseph R. Biden pledged to follow the terrorists “to the gates of hell” — but neither man offered a strategy or was willing to say the U.S. is at war.

Meanwhile, frustrations on Capitol Hill have reached the boiling point. Lawmakers are pleading with the president to acknowledge that the U.S. is in a de facto state of war with the Islamic State group and must act accordingly.

Some lawmakers and counterterrorism analysts say war has either started or is inevitable.

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“I believe ISIS has declared war on the United States by killing Americans — these two journalists,” Rep. Michael T. McCaul, Texas Republican, told Fox News on Wednesday. “I do believe they’re intent on striking the West, including Western Europe and the homeland.”

Analysts say the Islamic State, with its capacity to capture and hold territory, ability to take high-profile hostages and desire to launch attacks against the West, could represent an even greater threat than al Qaeda in the days before Sept. 11, 2001.

For that reason, it’s a virtual certainty that the U.S. eventually will have to declare war against the group, said Jens David Ohlin, a professor at Cornell University School of Law who specializes in counterterrorism.

“The U.S. has substantially degraded al Qaeda’s military capacity, but now the danger posed by ISIS is even more acute. ISIS is trying — and succeeding — to carve out a permanent territory,” he said. “This will allow them not only to implement the harshest form of Shariah law on the domestic population, but will also provide a safe haven from which to launch international attacks. War seems inevitable.”

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Speaking in Estonia on Wednesday, Mr. Obama said the U.S. objective is clear but offered no blueprint for meeting that objective, nor did he indicate that he is any closer to authorizing airstrikes against the Islamic State in Syria.

“The bottom line is this: Our objective is clear. That is to degrade and destroy ISIL so that it is no longer a threat not just to Iraq but also to the region and to the United States,” Mr. Obama said. “In order for us to accomplish that, the first phase has been to make sure we’ve got an Iraqi government in place and that we are blunting the momentum that ISIL was carrying out. And the airstrikes have done that. But now what we need to do is make sure we’ve got the regional strategy in place that can