We investigate the impact of ram pressure stripping due to the intracluster medium (ICM) on star-forming disk galaxies with a multi-phase interstellar medium (ISM) maintained by strong stellar feedback. We carry out radiation-hydrodynamics simulations of an isolated disk galaxy embedded in a 10^11 Msun dark matter halo with various ICM winds mimicking the cluster outskirts (moderate) and the central environment (strong). We find that both star formation quenching and triggering occur in ram pressure-stripped galaxies, depending on the strength of the winds. HI and H$_2$ in the outer galactic disk are significantly stripped in the presence of the moderate winds, whereas turbulent pressure provides support against ram pressure in the central region where star formation is active. Moderate ICM winds facilitate gas collapsing, increasing the total star formation rates by ~40% when the wind is oriented face-on or ~80% when it is edge-on. In contrast, strong winds rapidly blow away neutral and molecular hydrogen gas from the galaxy, suppressing the star formation by a factor of two within ~200 Myr. Dense gas clumps with N_H > 10 Msun pc^-2 are easily identified in extraplanar regions, but no significant young stellar populations are found in such clumps. In our attempts to enhance radiative cooling by adopting a colder ICM of T=10^6K, only a few additional stars are formed in the tail region, even if the amount of newly cooled gas increases by an order of magnitude.
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