Donald J. Trump on Monday invoked comparisons to the Cold War era in arguing that the United States must wage an unrelenting ideological fight if it is to defeat the Islamic State. He said he would temporarily suspend immigration from “the most dangerous and volatile regions of the world” and judge allies solely on their participation in America’s mission to root out Islamic terrorism.

In a speech at Youngstown State University in Ohio, a critical swing state where polls show him trailing Hillary Clinton, Mr. Trump combined old vows to seize Middle Eastern oil fields with the announcement of a series of new, if still vague, proposals to change America’s battlefield tactics.

“Just as we won the Cold War, in part by exposing the evils of communism and the virtues of free markets, so too must we take on the ideology of radical Islam,” he said.

He again tried to change his politically inflammatory approach to immigration, replacing his 2015 vow to bar Muslims from entering the United States with a new commitment to bar anyone from parts of the world where terrorism breeds. Once again, he did not name those countries, or say whether citizens of longtime allies where terrorists have plotted and executed attacks — Germany, France and Belgium among them — would be included.