The most relevant impact of the Korean War on American race relations from the choices given is:
The U.S. military integrated Black and White troops during the war.
While this statement reflects the broader context of race relations during the era of the Korean War, it is important to note that the war accelerated discussions about racial integration and civil rights, even though the military was not fully integrated until after the war. The experience of fighting alongside one another contributed to growing awareness and calls for equality, influencing the post-war Civil Rights Movement.
The other options do not accurately reflect the direct impact of the Korean War on race relations, as the war itself was a complex period where policies were undergoing significant changes regarding race and military integration.