We compute the complete nonlocal one-loop form factors F₁(□/Λ²) and F₂(□/Λ²,ξ) of the curvature-squared sector of the spectral action S = Tr f(D²/Λ²) for the full Standard Model particle content: 4 real scalars (Higgs), 45/2 Dirac-equivalent fermions (3 generations), and 12 gauge bosons (SU(3)×SU(2)×U(1)). Using the Barvinsky-Vilkovisky covariant perturbation theory and the Codello-Zanusso diagrammatic heat kernel, we derive closed-form results for each spin sector (0, 1/2, 1) in the Weyl basis and assemble the Standard Model totals. The local limits yield α_C = 13/120 for the Weyl-squared coefficient and α_R(ξ) = 2(ξ-1/6)² for the R² coefficient, where ξ is the Higgs non-minimal coupling. Both form factors are shown to be entire functions, ensuring no additional propagator poles beyond those of the classical theory. We derive the c₁/c₂ ratio, the scalar graviton decoupling condition at conformal coupling ξ = 1/6, and the UV asymptotic behavior. The form factors yield a modified Newtonian potential with calculable effective masses m₂ = Λ(60/13)^1/2 and m₀ = Λ/(6(ξ-1/6)²)^1/2, connecting the spectral action framework to solar-system phenomenology. All results are verified by independent multi-precision numerical evaluation.