Higher-dimensional theories of gravity predict that gravitational waves (GWs) can propagate into extra spatial dimensions, leading to modified amplitude damping over cosmological distances. Measurements of GW luminosity distances therefore provide a unique probe of the dimensionality of spacetime. In this work, we constrain higher-dimensional GW propagation using the dark-siren method with the Gravitational-Wave Transient Catalog 4.0 (GWTC-4). We adopt a phenomenological parameterization motivated by braneworld scenarios, in which deviations from General Relativity are characterized by the spacetime dimension number $D$ and a crossover scale $R_c$ governing the transition between four- and higher-dimensional gravity. We perform a hierarchical Bayesian analysis combining 141 compact binary coalescences from GWTC-4 with line-of-sight galaxy information from the GLADE+ catalog. For a prior $H_0 \in [65,77]\ {\rm km~s^{-1}Mpc^{-1}}$ and $\log(R_c/{\rm Mpc}) \in [2.7,4.0]$, we obtain $D={4.38}_{-1.01}^{+1.91}$ (68% credible interval). We also find that the inferred posterior distribution of $R_c$ accumulates near the upper prior boundary, indicating that the crossover scale remains poorly constrained by current observations. We further show that the inferred constraint on $D$ depends sensitively on the assumed prior range of $R_c$, which determines the characteristic distance scale at which deviations from General Relativity become significant. Our results provide the first GWTC-4 dark-siren constraints on higher-dimensional GW propagation and demonstrate that current observations remain consistent with four-dimensional General Relativity.