In short: no.

Black men did not "have to be given" civil rights to be drafted into earlier wars. In fact, a major historical reason for the lack of black servicemen was the racist assumption that black personnel would be less qualified in their combat roles than their white counterparts. When numbers were needed, however, things became very different very quickly:

When it came to the draft, however, there was a reversal in usual discriminatory policy. Draft boards were comprised entirely of white men. Although there were no specific segregation provisions outlined in the draft legislation, blacks were told to tear off one corner of their registration cards so they could easily be identified and inducted separately. Now instead of turning blacks away, the draft boards were doing all they could to bring them into service, southern draft boards in particular. One Georgia county exemption board discharged forty-four percent of white registrants on physical grounds and exempted only three percent of black registrants based on the same requirements. It was fairly common for southern postal workers to deliberately withhold the registration cards of eligible black men and have them arrested for being draft dodgers. African American men who owned their own farms and had families were often drafted before single white employees of large planters. Although comprising just ten percent of the entire United States population, blacks supplied thirteen percent of inductees.

(That was in the middle of World War I, by the way.)

Civil Rights had nothing to do with the government's needs in wartime, although the draft itself was a rallying cry for civil rights leaders, who noted the absurdity of forcing black men to go to war for a government that denied them equality at home. Muhammad Ali, of course, famously refused to fight in Vietnam, saying:

Why should they ask me to put on a uniform and go ten thousand miles from home and drop bombs and bullets on brown people in Vietnam while so-called Negro people in Louisville are treated like dogs and denied simple human rights?

But while this did energize many young people into joining the Civil Rights movement, it did not and could not change the fact that draft dodging was a felony punishable by five years in prison; upon declaring himself a conscientious objector, Ali was stripped of his boxing license and title the very same day, and could not find a single state that would license him to box. The jury took only 21 minutes to find him guilty of violating the Selective Services Act, though the Supreme Court later voided that decision on a technicality once popular opinion had turned against the war.

So, again - I don't know if this response is long enough for this subreddit, but in short, no; your professor is abjectly wrong. The Vietnam War helped move the Civil Rights movement along, no question, but the government did not simply "agree to pass Civil Rights" because of a need for manpower in Vietnam, and I believe the premise of such a claim completely trivializes the massive sacrifices and suffering that the movement's adherents had to endure to get as far as they did.