The persistent discrepancy between the proton charge radius measured from muonic hydrogen ($r_p \approx 0.84 \text{ fm}$) and electronic hydrogen ($r_p \approx 0.88 \text{ fm}$)—the "proton radius puzzle"—suggests a potential violation of lepton universality or an incomplete understanding of vacuum structure at short distances. In this Letter, we propose a mechanism of \textbf{nonlinear vacuum metric response}. By treating the vacuum as a dielectric-like medium with a finite elastic modulus coupled to electromagnetic energy density, we derive a conformal scaling factor for the local metric. The contraction of the effective metric is analytically derived as $\eta = \alpha \ln(m_\mu/m_e)$, where the coupling factor arises from the summation over vacuum polarization modes. This formula yields a radius shrinkage of \textbf{3.8906\%}, which agrees with the experimental discrepancy of \textbf{3.9115\%} to within \textbf{0.02\%}. We further predict a radius of \textbf{0.823 fm} for tauonic hydrogen.