"The Discovery of Incommensurability" by Kurt von Fritz [ http://www.jstor.org/stable/1969021 ] indicates that the early Greek mathematicians did not explicitly use the Fundamental Theorem to prove the irrationality of √2. The proof known to Aristotle ("the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate") uses a restricted version of the Fundamental Theorem, as explained in http://en.wikipedia.org/wiki/Quadratic_irrational
Apparently, the explicit use of the Fundamental Theorem is post-Gauss.