Moore attended Peabody Demonstration School, now known as University School of Nashville, and in 1935 graduated summa cum laude from Vanderbilt University, where he was a member of Phi Kappa Sigma. He earned his doctorate in Organic Chemistry from the University of Wisconsin–Madison in 1938. Moore then joined the staff of the Rockefeller Institute, later Rockefeller University, where he spent his entire professional career, with the exception of a period of government Service during World War II. He became Professor of Biochemistry in 1952.
In 1958 he and william H. Stein developed the first automated amino acid analyzer, which facilitated the determination of protein sequences. In 1959 Moore and Stein announced the first determination of the complete amino acid sequence of an enzyme, ribonuclease, work which was cited in the Nobel award. He never married.