I think you might be better off asking, “What was the effect of US military policy on the Civil Rights Movement (CRM)?” Here’s how I’d start answering that question.

First, one would expect that being drafted, trained, and sent to war together would give whites and blacks some common ground they might not otherwise have had. Generally, we might say that white and black veterans (remember, it’s just men at this point) would have fewer stereotypes of each other after the wars were finished, since first-hand experience can replace stereotypes. So a thesis would be that compulsory military service created a more conciliatory environment of trust, which would have made the CRM more palatable to the country.

Some things to remember here are that the military was only integrated (by executive order, not an act of congress) in 1947. Many blacks served during WWII, but they did so in their own units. Korea was the first war in which men of both colors fought side by side in large numbers. In Vietnam, certain policies placed the draft burden on poor Americans, so that a disproportionate number of blacks served alongside their poor white counterparts. Also keep in mind that after WWII, the draft continued even during peaceful periods, due to the need to counter the USSR. (As an aside, military service is no long compulsory, but blacks continue to serve in disproportionate numbers, especially in the enlisted ranks.)

Second, you might also expect that black men who had served in the military would expect certain changes on their return. Having fulfilled a duty of citizenship, they might be very dissatisfied at being denied the benefits of citizenship—that is, the equal protection of the laws. There might also be an accompanying sentiment among whites that blacks, through their service, had proved their worth and loyalty, and deserved federal protection from discriminatory state laws.

Consider that some blacks also saw service in the combat arms (infantry, armor), especially in Korea and Vietnam. When these men returned, they might have adopted a more militant stance to the civil rights problem. I’m interested to know if anyone has studied veteran participation in the Black Panther Party, but I know that we saw black veterans taking up arms to defend themselves in the 1921 Tulsa Race Riot.

You might conclude, having done some research, that participation in the military provided a background against which the Civil Rights Movement made more sense to white Americans than it would have in the 19th century.