I present Quantum Elastic Spacetime Theory (QuEST), a first-principles framework in which spacetime emerges as an elastic quantum medium supporting strain, stress, and wave excitations. Within this formalism, gravitational phenomena are recovered as collective excitations of the elastic field, without recourse to an external geometric postulate. By deriving the field equations from a symmetry-constrained elastic Lagrangian, we obtain a continuum limit that reproduces Newtonian gravity, General Relativity, and their known weak- and strong-field tests, while also yielding new predictions in unexplored regimes. We show that QuEST naturally produces black hole echo delays through logarithmic scaling with core radius, in agreement with recent gravitational-wave observations, and yields a stable, dynamically supported low-lunar orbitwith negligible Δv requirements. In the astrophysical domain, QuEST explains the flat rotation curves of galaxieswithout invoking dark matter halos, as demonstrated with NGC 2403 using SPARC data. In the cosmological regime, an elastic background with ρₑₗ ≃ 0.7 ρ₍cᵣᵢₜ₎ provides a late-time acceleration nearly indistinguishable from ΛCDM, fully consistent with Pantheon+ supernova data. This unification across strong-gravity, galactic, and cosmological scales highlights QuEST as a predictive, parameter-economical alternative to both ΛCDM and particle dark matter models. Our results suggest that gravitational echoes, galactic rotation curves, and cosmic acceleration may all be manifestations of the same underlying elastic substrate of spacetime. The work provides testable predictions for future gravitational-wave detections, galaxy surveys, and precision cosmology, and opens a pathway toward a fully elastic, quantum-consistent description of gravity.