The case for dropping the atomic bomb revolves around the idea that a commander does everything in his power (that is legal) to do away with unnecessary risks to the troops under his command. Since dropping the bomb was clearly not illegal (any more than huge carpet bombing raids were illegal), it was right for President Truman to use the bomb to save lives.
In August of 1945, it seemed clear that an invasion of Japan would be horribly costly in terms of American lives (to say nothing of Japanese lives). The fighting on Saipan, Iwo Jima, Okinawa, and other places showed that the Japanese would fight practically to the last man. It was reasonable to expect that they would resist even more fiercely on their own home islands. Therefore, an invasion (the most likely alternative to using the bomb) would have killed and wounded hundreds of thousands of Americans. In addition, it would surely have killed more Japanese than the atomic bomb did.
For these reasons, dropping the bomb was justified.