In this paper we compare the molecular gas depletion times and midplane hydrostatic pressure in turbulent, star-forming disk galaxies to internal properties of these galaxies. For this analysis we use 17 galaxies from the DYNAMO sample of nearby (z similar to 0.1) turbulent disks. We find a strong correlation, such that galaxies with lower molecular gas depletion time (t(dep)) have higher gas velocity dispersion (sigma). Within the scatter of our data, our observations are consistent with the prediction that t(dep) proportional to sigma(-1) made in theories of feedback-regulated star formation. We also show a strong, single power-law correlation between midplane pressure (P) and star formation rate surface density (Sigma(SFR)), which extends for 6 orders of magnitude in pressure. Disk galaxies with lower pressure are found to be roughly in agreement with theoretical predictions. However, in galaxies with high pressure we find P/Sigma(SFR) values that are significantly larger than theoretical predictions. Our observations could be explained with any of the following: (1) the correlation of Sigma(SFR)-P is significantly sublinear; (2) the momentum injected from star formation feedback (p(*)/m(*)) is not a single, universal value; or (3) alternate sources of pressure support are important in gas-rich disk galaxies. Finally, using published survey results, we find that our results are consistent with the cosmic evolution of t(dep)(z) and sigma(z). Our interpretation of these results is that the cosmic evolution of t(dep) may be regulated not just by the supply of gas but also by the internal regulation of star formation via feedback.