Internal Elasticity Cosmology (IEC) has unified dark energy and dark matter as coherent and topological excitations of a single quantum-coherent elastic scalar field $\phi$, representing vacuum displacement. In this work —Stage 5— we demonstrate that metric gravity itself emerges as the low-energy collective mode of said elastic medium. By treating spacetime as a quantum elastic solid with stiffness $\kappa$ and shear modulus $\mu$ derived from the coherence of $\phi$, we derive Einstein's equations from the long-wavelength hydrodynamics of the elastic vacuum, reproducing General Relativity exactly in the classical limit while predicting controlled quantum deviations at Planck scales. Black hole horizons acquire an elastic surface tension, resolving the information paradox through coherent memory states; the cosmological constant problem is resolved via the relaxation of vacuum stiffness; and primordial gravitational waves acquire a characteristic testable dispersion relation for LISA and future interferometers. Thus, IEC becomes the first fully unified theory of quantum gravity, dark energy, dark matter, and large-scale structure emerging from a single physical principle: the internal quantum elasticity of spacetime.