UNITED NATIONS — President Obama on Wednesday charted a muscular new course for the United States in a turbulent world, telling the United Nations General Assembly in a bluntly worded speech that the American military would work with allies to dismantle the Islamic State’s “network of death” and warning Russia that it would pay for its bullying of Ukraine.

Two days after ordering airstrikes on dozens of militant targets in Syria, Mr. Obama issued a fervent call to arms against the Islamic State — the once-reluctant warrior now apparently resolved to waging a twilight struggle against Islamic extremism for the remainder of his presidency.

“Today, I ask the world to join in this effort,” Mr. Obama said, seeking to buttress a global coalition that he said would train and equip troops to fight the group, also known as ISIL, starve it of financial resources, and halt the flow of foreign recruits to its ranks.

“Those who have joined ISIL should leave the battlefield while they can,” Mr. Obama said, foreshadowing the blows to come. “For we will not succumb to threats, and we will demonstrate that the future belongs to those who build, not those who destroy.” The brutality of the militants, he said, “forces us to look into the heart of darkness.”