We compare the observed turbulent pressure in molecular gas, $P_mathrm{turb}$, to the required pressure for the interstellar gas to stay in equilibrium in the gravitational potential of a galaxy, $P_mathrm{DE}$. To do this, we combine arcsecond resolution CO data from PHANGS-ALMA with multi-wavelength data that traces the atomic gas, stellar structure, and star formation rate (SFR) for 28 nearby star-forming galaxies. We find that $P_mathrm{turb}$ correlates with, but almost always exceeds the estimated $P_mathrm{DE}$ on kiloparsec scales. This indicates that the molecular gas is over-pressurized relative to the large-scale environment. We show that this over-pressurization can be explained by the clumpy nature of molecular gas; a revised estimate of $P_mathrm{DE}$ on cloud scales, which accounts for molecular gas self-gravity, external gravity, and ambient pressure, agrees well with the observed $P_mathrm{turb}$ in galaxy disks. We also find that molecular gas with cloud-scale ${P_mathrm{turb}}approx{P_mathrm{DE}}gtrsim{10^5,k_mathrm{B},mathrm{K,cm^{-3}}}$ in our sample is more likely to be self-gravitating, whereas gas at lower pressure appears more influenced by ambient pressure and/or external gravity. Furthermore, we show that the ratio between $P_mathrm{turb}$ and the observed SFR surface density, $Sigma_mathrm{SFR}$, 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 $Sigma_mathrm{SFR}$ and kpc-scale $P_mathrm{DE}$ 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 $P_mathrm{DE}$ reported in previous works.
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