Hi there. The only major wars i can think of up to 1960 is World War 1, World war 2, and the Korean war. I thought maybe he would be referring to the Chinese civil war which i guess would count as a major war in terms of casualties. But otherwise I'm clueless as to what this fourth major war which did not involve the united states was.
The three wars involving the United States were WWI, WWII, and the Korean War. The fourth war is the Russo-Japanese War of 1904-1905. The Russo-Japanese War was one that Eisenhower took a great deal of interest in as a young man, tracking the movements of forces based on radio reports using a map and pins. Though other wars in the 20th century were more devastating, such as the Chinese Civil War and Spanish Civil War, those were not technically among great powers, as they were fought internally with unofficial/proxy support from other nations. The French experience in Vietnam also would not count as that was a colonial war during Eisenhower's presidency, which he also viewed as an internal affair for the French, though he was willing to provide some level of support to the French in the conflict.