We present the relation between the star formation rate surface density, $Sigma_{rm SFR}$, and the hydrostatic mid-plane pressure, P$_{rm h}$, for 4260 star-forming regions of kpc size located in 96 galaxies included in the EDGE-CALIFA survey covering a wide range of stellar masses and morphologies. We find that these two parameters are tightly correlated, exhibiting smaller scatter and strong correlation in comparison to other star-forming scaling relations. A power-law, with a slightly sub-linear index, is a good representation of this relation. Locally, the residuals of this correlation show a significant anti-correlation with both the stellar age and metallicity whereas the total stellar mass may also play a secondary role in shaping the $Sigma_{rm SFR}$ - P$_{rm h}$ relation. For our sample of active star-forming regions (i.e., regions with large values of H$alpha$ equivalent width), we find that the effective feedback momentum per unit stellar mass ($p_ast/m_ast$),measured from the P$_{rm h}$ / $Sigma_{rm SFR}$ ratio increases with P$_{rm h}$. The median value of this ratio for all the sampled regions is larger than the expected momentum just from supernovae explosions. Morphology of the galaxies, including bars, does not seem to have a significant impact in the $Sigma_{rm SFR}$ - P$_{rm h}$ relation. Our analysis suggests that self regulation of the $Sigma_{rm SFR}$ at kpc scales comes mainly from momentum injection to the interstellar medium from supernovae explosions. However, other mechanism in disk galaxies may also play a significant role in shaping the $Sigma_{rm SFR}$ at local scales. Our results also suggest that P$_{rm h}$ can be considered as the main parameter that modulates star formation at kpc scales, rather than individual components of the baryonic mass.
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