We develop a microscopic scattering theory of the phase-dependent equilibrium mechanical response of a ballistic superconductor-graphene-superconductor Josephson junction. The calculation is based on an energy-dependent Dirac scattering matrix embedded in a normalized Matsubara determinant, so the reported force correction contains the complete Bogoliubov-de Gennes spectrum of the stated ideal model. At charge neutrality the full determinant approaches a closed-form evanescent-mode result proportional to Δ W/L², reaching 2ln 2 Δ W/(π L²) at phase difference π in the zero-temperature short-junction limit. Gate doping produces propagating channels and Fabry-Pérot structure, causing large oscillations and sign reversals of the phase-dependent force correction δ F=F(φ)-F(0). We distinguish the interband-to-intraband Andreev crossover, controlled by |μ|/Δ, from the evanescent-to-propagating crossover, controlled by |μ|L/(ℏ v_F). Exact real-energy subgap poles obtained from the same energy-dependent scattering matrix are used to diagnose specular/interband and retro/intraband character; these labels are not treated as separately measurable thermodynamic forces in the mixed regime. We also quantify the difference between the complete determinant result and the frozen-scattering short-junction approximation without identifying that difference with a pure continuum force. The resulting gate- and phase-dependent mechanical signal provides a Dirac-material extension of earlier superconductivity-induced mechanical-force proposals.