I doubt that the dropping had any real effect on Claude Eatherly's life at all. Except for the fact that he was not the one to do it. Maybe if he had dropped the atomic bomb on Hiroshima, he would have found the fame he sought and better adjusted to civilian life.
Back in the 60's he was the poster boy for the "Ban the bomb" crowd. The put Eatherly on a pedestal and said "Look what this horrible incident has done to this man!" What they did not do was delve deeper into Eatherly's past. If they had, they would have seen that he was no different then he had been before.
Both before and during his time in the 509th, he was a heavy drinker, a heavy gambler, and a womanizer. His antics caused Tibbet's many problems, but Tibbets never got rid of him because of Eatherly's abilities as a pilot. During his time, stateside with the 509th, Eatherly had many run-ins with the law, Tibbets would keep him out of jail, but force him to pay any fines.
Not only was Eatherly a playboy, but he was a glory-seeker as well. During one of the 509ths "Pumpkin" missions, after finding his original target obscured by clouds, Eatherly wanted to drop his on the Imperial Palace in Tokyo. But, when the "Straight Flush" reached Tokyo, the city was also covered by clouds, so they dropped by radar. Upon returning to base, they listened intently to "Tokyo Rose" for any mention of their attack, but none were fourthcoming. The bomb had fallen well away from its intended target.
Still, for all the man's faults he was one hell of a B-29 pilot. Unfortunately, when the war ended, his abilities as a pilot were not enough to provide him with the lifestyle he so craved.