Nolan mentioned the indigenous population of New Mexico a few times, but without details. Obviously the US, Soviet Union and to the lesser extent the UK and France did a lot of terrible things to native people of various regions in the world, just to test the A bombs... Here's how they were treating native people from the Bikini Islands: 
However, I don't think that Oppenheimer was the war criminal himself. When it comes to the unfair treatment of New Mexico residents, it's the US Army to blame. Yes, the atomic bomb has killed about 220-250 000 people, however it also prevented a lot of big wars, in which the loss of life would be far, far greater in numbers.
Had there been no bomb, millions of people would have died during the invasion of Japan. And the Soviets, more likely then not, would have launched a ground invasion of Europe in late 40's or early 50's...
RE: "Oppenheimer" - A Historical Biopic that Falls Short of Nuclear Brilliance