We derive the modified Newtonian potential and post-Newtonian parameters of the spectral action with full Standard Model content. The linearized field equations yield two parameter-free Yukawa corrections with amplitudes −4/3 (spin-2) and +1/3 (spin-0) and calculable effective masses m₂ = Λ√(60/13) and m₀ = Λ/√(6(ξ−1/6)²). We confront these predictions with seven gravitational experiments. The Eöt-Wash torsion-balance experiment provides the strongest constraint: Λ > 2.565 × 10⁻³ eV (95% CL), corresponding to 1/Λ < 77 μm. All solar-system tests are satisfied with exponential margin. Companion paper to "Nonlocal one-loop form factors of the spectral action with Standard Model content" (doi:10.5281/zenodo.19039242).