Barely had I been seated before Vladimir Putin told me that NATO—the organization that I then headed—no longer had any purpose and should be disbanded. “After the end of the Cold War, we dissolved the Warsaw Pact,” he said. “Similarly, you should dissolve NATO. That is a relic from the Cold War.”

During my visit to Moscow in December 2009, I sensed that President Putin was challenging the world order that the U.S. created so successfully after World War II. Beginning in 2014, he invaded Ukraine and launched a military action...