We propose a theoretical cosmology framework in which the classical spacetime manifold is reinterpreted as an emergent superfluid vacuum, described by a Bose--Einstein condensate governed by a nonlinear textit{logarithmic Schr"{o}dinger equation} (LogSE). In this two-phase picture, the homogeneous ground-state of the condensate (Phase A) gives rise to cosmic acceleration (dark energy) through its negative pressure and exhibits a small bulk viscosity that can reconcile disparate measurements of the Hubble constant. Meanwhile, excited states of the condensate (Phase B) form quantized vortices and density solitons that behave as dark matter halos in galaxies. We derive the effective fluid dynamics of this superfluid vacuum, showing that it naturally yields a cosmic equation-of-state $w approx -1$ on large scales and MOND-like phenomenology on galactic scales, without requiring unknown particle species. We demonstrate that quantum pressure from the LogSE resolves the core--cusp problem by stabilizing galactic cores, and that the logarithmic self-interaction allows halo core sizes to be decoupled from the particle mass, avoiding the Catch-22'' that plagues fuzzy dark matter. The framework is confronted with observations: it passes current cosmological tests and galactic rotation curve data, while making distinct, falsifiable predictions. In particular, Lorentz invariance emerges only as a low-energy symmetry of the superfluid vacuum, implying an energy-dependent vacuum refractive index at high energies. We discuss how precision multimessenger timing (e.g., GW170817) and ultra-high-energy gamma-ray observations (e.g., LHAASO detection of GRB~221009A) place stringent constraints on any such Lorentz-violating dispersion. Upcoming astronomical surveys and particle experiments will further test this unified dark'' sector framework.