The Gauss-Newton method can be derived from Newton's method, but I am unable to see how Gauss was linked with this method. It seems unlikely that he himself worked on the method, but I am at a loss.
Undoubtedly, Gauss' name is attached to the name of the Gauss-Newton method due to his pioneering role in the development and application of the least-squares method to non-linear least-squared problems. As to whether he actually conceived of and employed the Gauss-Newton method to solve such problems, the paper "A Gauss-Newton method for convex composite optimization” makes that claim in the introduction.
Addendum: After some more searching, I found where Gauss published his variant of Newton's method: sections 179-180 of his Theoria motus corporum coelestium in sectionibus conicis solem ambientum (1809).