We study the magnetic and electronic phases of a one-dimensional (1D) magnetic adatom chain on a 2D superconductor. In particular, we confirm the existence of a "self-organized" 1D topologically nontrivial superconducting phase within the set of subgap Yu-Shiba-Rusinov states formed along the magnetic chain. This phase is stabilized by incommensurate spiral correlations within the magnetic chain that arise from the competition between short-range ferromagnetic and long-range antiferromagnetic electron-induced exchange interactions, similar to a recent study for a 3D superconductor [M. Schecter, Phys. Rev. B 93, 140503(R) (2016)1098-012110.1103/PhysRevB.93.140503]. The exchange interactions along diagonal directions are also considered and found to display behavior similar to a 1D substrate when close to half filling. We show that the topological phase diagram is robust against local superconducting order parameter suppression and weak substrate spin-orbit coupling. Lastly, we study the effect of a direct ferromagnetic exchange coupling between the adatoms, and find the region of spiral order in the phase diagram to be significantly enlarged in a wide range of the direct exchange coupling.