Evidence

     Dropping the atomic bomb on Hiroshima was necessary because if the U.S. military hadn't dropped it, a bloody invasion of the Japanese Home Islands would have taken place, where they would have had defender's advantage and been able to counter our superior naval, land and air firepower.  Also, if we didn't drop the bomb, Japan had an extensive and lethal biological warfare program that could have killed millions of American and Allied troops.
     If you look at it from the point of view of the Japanese, the bomb was considered unnecessary; but if you looked at it from a military personnel's point of view, the bomb was our only option to end the war quickly.  The Japanese nature of war is to fight to the death and surrender wasn't an option; they would rather kill themselves than surrender.  It was considered shameful to return from war defeated.  With this strategy, the war against Japan would last years and cost millions of lives.
     Dropping the bomb also justified the torturious war strategies of Japan.  Japan had killed 370,000 Chinese in the Rape of Nanking, 2,400 U.S. military and civilians in the bombing of Pearl Harbor, and between 6,000 and 11,000 Filipinos and Americans in the Bataan Death March.  If the bomb hadn't been dropped, the Japanese would have killed countless other prisoners of war and troops from other countries.
    

American Military History: Position Paper No. 5
Secretary of War Henry Stimson thought that "the bomb should be used to show the world the destruction of war so that the world could achieve everlasting peace".  Arguments within the government have varied from wanting to get even with Japan for Pearl Harbor to Congress's justification of the spending of $2 billion to create the two atomic bombs.  American historian and biographer Stephen Ambrose wrote in his book Rise to Globalism that "the killing of a few more 'Japs' seemed the natural thing to do to get them to surrender."  "The answer to why the United States dropped the atomic bomb on Japan is simple; the United States had the bomb, Japan's military was dedicated, and Japan had not surrendered".  Japan had their chance to surrender before the U.S. dropped the bomb, but the Japanese Premier Suzuki said the the ultimatum demanding Japan's surrender was "not worthy of public notice".
In a 1955 interview, President Truman justified his reasons for using such a deadly weapon,
"It was the estimate of General [George] Marshall that this action [dropping the bombs] would probably save the lives of 250,000 of our soldiers and probably twice as many casualties -- that was what I was trying to avoid...
...In the long run it takes the psychological condition of the enemy's mind to cause a surrender but my objective was to use the atomic bomb pirely as a military blow to create military surrender.  That in the long run is what happened.  In the first World War if you will remember the Germans were not completely defeated -- Germany was never invaded and never touched except by a few air raids which we made along toward the end, but the fact that nine hundred thousand Americans had marched up to and were about to take over caused them to surrender.  they never would have surrendered otherwise.  I don't believe in speculating on the mental feeling and as far as the bomb is concerned I ordered its use for a military reason -- for no other cause -- and it saved the lives of a great many of out soldiers.  That is all I had in mind."
Pilot of Plane That Bombed Hiroshima Dies
Paul Tibbets, who piloted the plane that dropped the atomic bomb on Hiroshima, says he has no regrets of dropping the bomb.  Paul said in a 1975 interview, “I’m not proud that I killed 80,000 people, but I’m proud that I was able to start with nothing, plan it and have it work as perfectly as it did”.

Hiroshima: The Harry Truman Diary and Papers
In a letter to his wife, Truman wrote, "...I've gotten what I came for - Stalin goes to war August 15 with no strings on it. He wanted a Chinese settlement - and it is practically made - in a better form than I expected. Soong did better than I asked him. I'll say that we'll end the war a year sooner now, and think of the kids who won't be killed! That is the important thing".
In a later letter to Samuel McCrea Cavert, he wrote, "Nobody is more disturbed over the use of Atomic bombs than I am but I was greatly disturbed over the unwarranted attack by the Japanese on Pearl Harbor and their murder of our prisoners of war. The only language they seem to understand is the one we have been using to bombard them... When you have to deal with a beast you have to treat him as a beast. It is most regrettable but nevertheless true."

Interim Commitee Diray Entries
On page 2 of the diary entries, the Interim Commitee reccomends that "the bomb should be used against Japan, without specific warning, as soon as possible, and against such a target as would make clear its devastating strength."  Not only did the Interim Commitee agree with the use of the atomic bomb against Japan, but when Truman told Stalin that the U.S. had developed a nuclear weapon, Stalin showed "no special interest and that the Generalissimo did not seem to have any conception of what Truman was talking about."

Draft Statement on the Dropping of the Bomb, July 30, 1945; President Secretary's File, Truman Papers
In these papers, on pages 2-3, it explains the reasons why Churchill and Roosevelt agreed that it was necessary to use the bomb. "  "The United States had available the large number of scientists of disctinction in the many needed areas of knowledge.  It had the tremendous industrial and financial resources necessary for the project and they could be devoted to it without undue inpairment of other vital war work.  In the United States the laboratory work and the production plants, on which a substantial start had already been made, would be out of reach of enemy bombing, while at that time Britain was exposed to constant air attack and was still threatened with the possibility of invasion."

The Bataan Death March - A Survivor's Story
Part of the reason the U.S. dropped the bomb on Japan was because Japan had tortured and killed thousands of prisoners of war during the Bataan Death March.  In this interview with a survivor of the march, he describes what the Japanese soldiers did to the P.O.W's.  "If people would fall down and couldn't go any further, the Japanese would either bayonet or shoot them. They also would bayonet prisoners who couldn't keep up...
Those who stepped out of line or had fallen out of ranks were beaten with clubs and/or rifle butts. Some American prisoners who couldn't keep up were run over by Japanese vehicles. I saw the remains of an American soldier who had been run over by a tank. I didn't see the actual event but the Japanese just left his remains in the middle of the road."

Press Release by the White House, August 6, 1945. Subject File, Ayers Papers.
This document was sent out the day the atomic bomb was dropped.  It explains the justification of why it was necessary to drop the bomb on Japan.  "It was to spare the Japanese people from utter destruction that the ultimatum of July 26 was issued at Potsdam.  Their leaders promptly rejected that ultimatum.  If they do not now accept our terms they may expect a rain of ruin  from the air, the like of which has never been seen on this earth.  Behind this air attack will follow sea and land forces in such numbers and power as they have not yet seen and with the fighting skill of which they are already well aware."

Press Release by Henry Stimson, August 6, 1945. Subject File, Ayers Papers.
This press release by Stimson was released to show the incredible advancement in science and technology that the U.S. made by creating and dropping the atomic bomb.  "The recent use of the atomic bomb over Japan, which was today made known by the President, is the cumulation of years of herculean effort on the part of science and industry working in cooperation with the military authorities.  this development which was carried forward by the many thousand participants with the utmost energy and the very highest sense of national duty, with the greatest secrecy and the most imperative of time schedules, probably represents the greatest achievement of the combined efforts of science, industry, labor, and the military in all history."