Who Army bases are named after is not really relevant to racism in the military in a practical sense - less so if you're interested in being a Naval Officer.
A better question is to understand the demographics in the Officer Corps to explain the demographics of those in "top jobs." Generally speaking, minorities of all kinds are overrepresented in the enlisted ranks, with respect to their numbers in the US population. In the officer ranks, minorities, and especially women, are underrepresented. Lots of factors the go into why that is, but we'll set those aside for now. With minorities being a very small subset of all officers - usually less than 10%, it's natural that there will be fewer eligible minority officers competing for those "top jobs." In many respects, it's just a numbers game. The same holds true for women, who are around 5-7% of the officer corps in the USN.
I would say that our promotions process for officers is at least as fair as anything in the civilian world. Some would argue that there are even mechanisms that look like affirmative action to give minority officers (and women) some advantages in reaching senior positions. That is debatable, but it's an idea that lingers.
So, my perspective (as a white male, granted) is that the military is, on balance, less concerned about someone's race than most large organizations in the corporate world. If you do good work, you will generally be rewarded. The Navy is working hard to change the demographics of the people it hires to more closely match the population at large - more women, more racial minorities, etc. That takes time.