We formulate the GSM in coupled-channel (GSM-CC) representation to describe low-energy elastic and inelastic scattering of protons on $^{18}$Ne. The GSM-CC formalism is applied to a translationally-invariant Hamiltonian with an effective finite-range two-body interaction. We discuss in details the GSM-CC formalism in coordinate space and give the description of the novel equivalent potential method for solving the GSM-CC system of integro-differential equations. We present the first application of the GSM-CC formalism for the calculation of excited states of $^{18}$Ne and $^{19}$Na, excitation function and the elastic/inelastic differential cross-sections in the $^{18}$Ne$(p,p')$ reaction at different energies.