We present a rigorous construction and uniqueness proof of the matrix Green's function for coupled radial Schrödinger equations with symmetric coupling potentials. The Green's matrix $g_{γγ'}(R,R')$ is built from two fundamental sets of $N$ linearly independent solutions, regular and outgoing, of the coupled radial equations. We prove that the associated Wronskian matrix is diagonal with elements $W_n = -k_n$ and independent of the radial coordinate, and demonstrate through the symplectic structure of the $2N$-dimensional phase space that the resulting construction is the unique Green's matrix satisfying the defining equation with correct boundary conditions, continuity at the source point, and the prescribed derivative discontinuity. The construction applies to any system of coupled radial Schrödinger equations with symmetric coupling potentials and open channels, including coupled-channels problems arising in nuclear, atomic, and molecular scattering. As an illustrative application, we show how the Green's matrix enters the nonlocal dynamical polarization potential (DPP) within the continuum-discretized coupled-channels (CDCC) framework, where retaining the off-diagonal elements captures multistep excitation pathways beyond the weak-coupling approximation.