We compare the observed turbulent pressure in molecular gas, P_turb_, to the required pressure for the interstellar gas to stay in equilibrium in the gravitational potential of a galaxy, P_DE_. To do this, we combine arcsecond resolution CO data from PHANGS-ALMA with multiwavelength data that trace the atomic gas, stellar structure, and star formation rate (SFR) for 28 nearby star-forming galaxies. We find that P_turb_ correlates with--but almost always exceeds--the estimated P_DE_ on kiloparsec scales. This indicates that the molecular gas is overpressurized relative to the large-scale environment. We show that this overpressurization can be explained by the clumpy nature of molecular gas; a revised estimate of P_DE_ on cloud scales, which accounts for molecular gas self-gravity, external gravity, and ambient pressure, agrees well with the observed P_turb_ in galaxy disks. We also find that molecular gas with cloud-scale P_turb_~P_DE_>~10^5^k_B_Kcm^-3^ in our sample is more likely to be self-gravitating, whereas gas at lower pressure it appears more influenced by ambient pressure and/or external gravity. Furthermore, we show that the ratio between P_turb_ and the observed SFR surface density, {Sigma}_SFR_, is compatible with stellar feedback-driven momentum injection in most cases, while a subset of the regions may show evidence of turbulence driven by additional sources. The correlation between {Sigma}_SFR_ and kpc-scale P_DE_ in galaxy disks is consistent with the expectation from self-regulated star formation models. Finally, we confirm the empirical correlation between molecular-to-atomic gas ratio and kpc-scale P_DE_ reported in previous works.