The quenching of cluster satellite galaxies is inextricably linked to the suppression of their cold interstellar medium (ISM) by environmental mechanisms. While the removal of neutral atomic hydrogen (H i) at large radii is well studied, how the environment impacts the remaining gas in the centres of galaxies, which are dominated by molecular gas, is less clear. Using new observations from the Virgo Environment traced in CO survey (VERTICO) and archival H i data, we study the H i and molecular gas within the optical discs of Virgo cluster galaxies on 1.2-kpc scales with spatially resolved scaling relations between stellar ( $\Sigma_{\star}$ ), H i ( $\Sigma_{\text{H}\,{\small\text{I}}}$ ), and molecular gas ( $\Sigma_{\text{mol}}$ ) surface densities. Adopting H i deficiency as a measure of environmental impact, we find evidence that, in addition to removing the H i at large radii, the cluster processes also lower the average $\Sigma_{\text{H}\,{\small\text{I}}}$ of the remaining gas even in the central $1.2\,$ kpc. The impact on molecular gas is comparatively weaker than on the H i, and we show that the lower $\Sigma_{\text{mol}}$ gas is removed first. In the most H i-deficient galaxies, however, we find evidence that environmental processes reduce the typical $\Sigma_{\text{mol}}$ of the remaining gas by nearly a factor of 3. We find no evidence for environment-driven elevation of $\Sigma_{\text{H}\,{\small\text{I}}}$ or $\Sigma_{\text{mol}}$ in H i-deficient galaxies. Using the ratio of $\Sigma_{\text{mol}}$ -to- $\Sigma_{\text{H}\,{\small\text{I}}}$ in individual regions, we show that changes in the ISM physical conditions, estimated using the total gas surface density and midplane hydrostatic pressure, cannot explain the observed reduction in molecular gas content. Instead, we suggest that direct stripping of the molecular gas is required to explain our results.