We have extended the fractional-dimensional space approach to study exciton states and diamagnetic shifts in symmetric coupled double GaAs-Ga_1-xAl_xAs quantum wells. In this scheme, the fractional dimension is chosen using an analytical procedure, and the real anisotropic `exciton + double quantum well' semiconductor system is mapped, for each exciton state, into an effective fractional-dimensional isotropic environment. We have performed calculations within the fractional-dimensional space scheme for the binding energies of 1s-like heavy-hole direct excitons and for the energy difference between 1s- and 2s-like direct heavy-hole exciton states in GaAs-Ga_1-xAl_xAs symmetric coupled double quantum wells. Also, theoretical results were obtained for the magnetic-field dependence of the 1s-like heavy-hole exciton energy shift and for the exciton diamagnetic coefficient in quantum wells and symmetric coupled double quantum wells. Fractional-dimensional theoretical results are shown to be in good agreement with available experimental measurements and previous theoretical calculations.