The dimensionless tidal deformability, $\Lambda$, of neutron stars (NSs), inferred from gravitational-wave (GW) observations, has thus far been used primarily to constrain the pressure of dense matter near twice nuclear saturation density, leaving the core equation of state (EOS) largely inaccessible to inspiral-phase GW observations. We show that the a fundamental central EOS parameter can be probed directly through $\Lambda$ using a perturbative analysis of the dimensionless stellar-structure and tidal-response equations formulated in terms of scaled intrinsic variables, without invoking any specific EOS model. We uncover a remarkable EOS-insensitive scaling relation between $\Lambda$ and the central EOS parameter $\x\equiv P_{\rm c}/\varepsilon_{\rm c}$, where $P_{\rm c}$ and $\varepsilon_{\rm c}$ denote the central pressure and energy density, respectively. The relation is validated against a broad ensemble of physically viable EOSs. Applying it to tidal deformabilities inferred from events such as GW170817 enables a direct determination of $\x$. By adopting the previously established effective upper bound $\x\lesssim0.381$, we infer a tight lower bound, $\Lambda_{\rm{TOV}}\gtrsim9.1^{+0.5}_{-0.5}$, for maximum-mass NSs along stable mass-radius sequences, quantitatively demonstrating that even the most compact stable NSs remain distinctly separated from black holes, for which $\Lambda_{\rm{BH}}=0$. These findings reveal a previously unrecognized connection between inspiral-phase tidal deformability and the central EOS parameter $\x$, establishing a direct link between GW observables and the microphysics of ultradense matter in the strong-gravity regime. The resulting scaling establishes inspiral-phase tidal deformability as a direct and largely model-insensitive probe of the central EOS parameter $\x$, and hence of key properties of NS cores.