What evidence could a prosecutor use "against" Stalin to prove that he was mostly responsible for the the bombs on Hiroshima/Nagasaki?
Arguing in favor of Stalin’s responsibility for United States dropping atomic bombs on the Japanese cities of Hiroshima and Nagasaki is a tricky issue, specifically because Stalin’s responsibility in the matter, insofar as it can be determined, does not rest in his direct involvement in the events. That is not to say that Stalin is not at least partially responsible for the actions of the United States against Japan in August 1945.
The most revealing factor in this equation is the generally uneasy relationship between the Soviet Union and the United States during World War II. Since Russia had pulled out of World War I after the October Revolution in 1917, the United States was cautious in its dealings with the Communist government that had replaced the Czar. This uneasiness and caution was exacerbated by Stalin’s actions near the end of World War II. As the Allies began to retake those lands to which Germany had laid claim in the years leading up to the war, the Soviet Union seized the opportunity to create satellite states from the countries of Eastern Europe. The United States, perceiving Stalin’s actions to be a repeat of Hitler’s aggression, feared that Communism would spread throughout the rest of Europe. It has been argued that this fear prompted Harry S. Truman to drop the bombs on Hiroshima and Nagasaki. Detonating the bombs was a show of force meant to send Stalin a message: if you keep doing what you are doing, this could happen to you! The Soviet Union being the central source for American fears prompted the United States to commit these two acts.
A less compelling argument for what transpired is that the Soviet Union failed to commit to armed action against Japan until Germany was defeated. The United States, forced to fight on separate fronts and not receiving adequate military aid from the other Allies, could not wait for the conclusion of affairs in Europe; the loss of life in the fight to take Japan had simply taken too much of a toll. Fearing that they could not hold out for Soviet aid, Truman perceived the use of the atomic bomb as the only viable option left open to him. If the Soviet Union had committed to aiding the American campaign against Japan, the use of the atomic bombs may not have proven necessary. While Stalin did not push the button to drop the bombs, his actions, or perhaps his lack of action, certainly affected American military action against Japan.
I think it would be difficult to make a case for this, although I can see some minor sense to it. Yes, the lack of action against Japan by the USSR meant that America bore the brunt of the Pacific War, and yes the Russians had their hands full with Germany. Still, Stalin had plenty of troops available for massive relocation and military action against the Chechens and other ethnic groups in the Caucases during the war, peoples which had not helped the Germans at all and who had supported the Red Army during the Civil War following the Revolution.
It was obvious that any Soviet help fighting Japan would result in Russian domination of any territory they might take (they have never returned the Kuriles to Japan). But the reason for the atomic bombing of the two cities was a purely unilateral military decision to bring the biggest available "gun" to bear in an attempt to force a decision before an actual and extremely costly invasion of Japan. The Japanese military planned total resistance with the entire population of the country to invasion, and if defeated on their native soil to withdraw the government to the mainland of China, where they still had over a million troops with all their equipment and held well over half of China, plus Manchuria, Korea and Indochina. The US could never have expected any aid from Russia in that theatre of war. I do agree that if Stalin bears any responsibility it was through inaction rather than action, but I believe he had every confidence America would prevail eventually, plus the hope that the war would drag on for a few more years, destroy Japan and weaken the US so badly that he would then have a free hand in Asia. Although there were people aware of the bomb who might have thought it would send a message to Stalin, I believe Truman's decision was purely directed at the Japanese militarist government.