Vietnam became central to the civil rights movement. For one thing, although the military was fully desegregated, institutionalized racism persisted. I'm going to quote from a thread I participated in a few months ago on this topic:

Race relations were indeed extremely fraught during the Vietnam War. It was really the first war in which the military was fully desegregated (Truman ordered the military desegregated in 1948, but it didn't happen in practice until 1951), and many whites, especially career NCOs, had a hard time dealing with that. Many of those white NCOs were from the south, which sometimes escalated matters. For instance, there were a number of race brawls the day that Martin Luther King was killed, and one of the biggest took place on a base where someone had the bright idea to raise the Confederate flag.

Most of the incidents of racial violence within Vietnam were alcohol-fueled brawls. It was reasonably common for fights to break out in officers' clubs and other areas where alcohol was served. Fragging (troops murdering their superiors) usually did not happen for race related reasons. Scholars tend to agree that while there are a few isolated instances of racially-motivated fraggings, these incidents typically took place because of generally poor morale. Soldiers might frag their superiors for keeping them out in the jungle too long, or for threatening to send them to the brig, rather than race-related reasons. We don't know about fraggings in the field, however; there are no records for that. Besides individual fights, major race riots did occasionally take place, such as the Long Binh Jail riot of 1968, in which several hundred black inmates rioted, caused major damage to the base, and injured several dozen white inmates and guards.

The Vietnam-era draft process certainly had some racist features, but its most egregious problems were class-based. One could receive a deferment from the draft by maintaining enrollment in college or graduate school. Hence, most of the troops sent to Vietnam, white or black, were poor or working class kids who had flunked/dropped out of college at best. African-American participation in the war was high. One of the major reasons for that was military recruiters' skill at using draft avoidance as a selling point for enlisting. Recruiters would often pitch enlisting as a way for some kid to get a better job than he would if he waited to be drafted. The line was essentially "You're going no matter what you do, so you might as well enlist rather than wait for the draft board."

For the first few years of the war, African-American death rates were disproportionately higher than white death rates. There are a variety of theories as to why this happened, ranging from racist white NCOs placing blacks up front to black troops volunteering for combat duty to prove their valor and worth as men. Regardless, the high death rate, when combined with poor conditions for African-Americans on the home front, helped make Vietnam a galvanizing force for the civil rights movement by the mid-1960s. Despite his friendly relationship with Lyndon Johnson, Martin Luther King began to speak out against the war in 1965.

Vietnam also became a driving force for the more radical members of the civil rights movement. Many black nationalists and Black Power advocates saw the war as a clear example of American imperialism. Stokely Carmichael is frequently credited with saying that the war was "the white man forcing the black man to make war on the yellow man to defend the land he stole from the red man." Solidarity with the developing world (or Third World) was a big part of Black Power, since many BP activists saw African Americans as oppressed people along the same lines as people in colonized areas. Indeed, many BP activists often referred to American ghettos as "internal colonies." Black nationalists therefore encouraged solidarity with international anti-colonial revolutions.

Black Power also became widespread within the military. Again, to quote from my previous comment:

Blacks would often exchange the black power salute instead of the military salute, and "dapping" or complicated handshakes meant to express racial solidarity, became pretty frequent as well. Black troops would also carry ebony canes, and the Army actually started offering black hair products and dashikis in PXs throughout Vietnam at one point in an attempt to soothe racial tensions. The Navy, which was probably the most racially retrograde of the armed services during the war, had to deal with two major instances of racial unrest aboard the aircraft carriers Kitty Hawk and Constitution in 1971. In both cases blacks rioted in response to how racist the Navy's command structure still was. Black Power played a big role in these riots, and many of the rioters were photographed giving Black Power salutes. Thankfully, the Chief of Naval Operations at the time (Adm. Elmo "Bud" Zumwalt) was committed to reforming the Navy, and used the riots as a jumping off point for that.

Overall, Vietnam was very important for the civil rights movement, and vice versa. The civil rights movement helped to expose many racial problems within the military. And the war helped to give focus and direction to many social change activists.

EDIT to provide sources:

Anderson, Terry. The Movement and the Sixties: Protest in America From Greensboro to Wounded Knee. Oxford, 1995.

Appy, Christian. Working Class War: American Combat Soldiers and Vietnam. UNC Press, 1993.

Joseph, Peniel E. (editor). The Black Power Movement. Taylor and Francis, 2006.

Phillips, Kimberley L. War! What Is It Good For? Black Freedom Struggles From World War II to Iraq. UNC Press, 2014.

Westheider, James E. Fighting on Two Fronts: African Americans and the Vietnam War. NYU Press, 1997.