Gravity Screening from Matter Density: Resolving the Cosmological Constant and UV Divergences without New Degrees of Freedom

We propose a modification of General Relativity in which the gravitational coupling depends on the local matter energy density ρ_m through a screening function F(ρ_m) = 1 - exp(-ρ_m/ρ_c), with ρ_c the Planck density. In regions where ρ_m → 0, gravity decouples from the vacuum energy, providing a natural suppression of the cosmological constant by 120 orders of magnitude without fine-tuning. At high densities ρ_m → ρ_c, the coupling saturates, introducing an effective UV cutoff at the Planck scale that renders gravitational loop integrals finite. The theory contains no new propagating degrees of freedom and reduces to standard GR for ρ_m ≪ ρ_c. We derive three testable predictions: (1) a shift in the f-mode frequency of neutron star mergers at the 10^-10 level, detectable with Einstein Telescope; (2) a 2.7% deviation in the expansion history H(z) at z = 2-4, accessible to Euclid and DESI; and (3) a sharp cutoff in the gamma-ray spectrum from primordial black hole evaporation at 10^19 GeV. The model is consistent with Solar System tests and avoids ghosts and Jeans instabilities at linear order. This framework offers a minimal, falsifiable resolution to the cosmological constant problem and the UV divergences of quantum gravity.