This work presents the first full observational validation of the RMB (Raum–Materie–Bewegung, space–matter–motion) cosmological framework. The model is based on the covariant motion–space vector field Φ_μ = f(H) u_μ, with f(H) = 1/(E² + 1) and E = H/H₀, introduced in RMB Cosmology I. We derive and implement the unique implicit Friedmann equation governing the screened RMB expansion history, including the background energy density ρ_RMB(H) = 4 Ω_RMB,0 ρ_crit,0 f²(H), leading to the implicit cubic equation for the expansion function E(z). A full numerical analysis is performed, covering cosmic time, distance measures, the luminosity–distance relation, and two-dimensional likelihood maps in (H₀, Ω_m). All computations are validated using independent solvers with a precision of better than 10⁻⁶. Using the Pantheon+ Type Ia supernova compilation (1300 objects with full covariance) and BAO measurements from BOSS DR12 and eBOSS DR16, we find that the screened RMB model yields significantly improved supernova fits compared to ΛCDM: Supernova fit quality: χ²_SN/N = 1.84 (RMB) vs 3.84 (ΛCDM) → Improvement: Δχ²_SN = −2599 Combined Pantheon+ + BAO likelihood: χ²_total/N = 2.72 (RMB) vs 4.29 (ΛCDM) → Improvement: Δχ²_total = −2065 Two parameter regimes emerge: (1) a shallow interior minimum at moderate (H₀, Ω_m), and (2) a boundary minimum at higher values where the RMB screening term dominates. The global best-fit found within the scanned range is H₀ ≈ 102.6 km s⁻¹ Mpc⁻¹, Ω_m ≈ 0.527, χ²_min ≈ 1901.7. The reduced sound horizon predicted by the model, r_s ≈ 94–123 Mpc, naturally produces BAO tensions, confirming the need for early-Universe refinements in RMB Cosmology III. This preprint establishes the validated late-time behaviour of the RMB cosmology and provides a mathematically controlled and observationally competitive alternative to ΛCDM without invoking dark energy or additional scalar fields.