© 2016. The American Astronomical Society. All rights reserved. We show for the first time that H2 formation on dust grains can be enhanced in disk galaxies under strong ram pressure (RP). We numerically investigate how the time evolution of H i and H2 components in disk galaxies orbiting a group/cluster of galaxies can be influenced by the hydrodynamical interaction between the gaseous components of the galaxies and the hot intracluster medium. We find that compression of H i caused by RP increases H2 formation in disk galaxies before RP rapidly strips H i, cutting off the fuel supply and causing a drop in H2 density. We also find that the level of this H2 formation enhancement in a disk galaxy under RP depends on the mass of its host cluster dark matter halo, the initial positions and velocities of the disk galaxy, and the disk inclination angle with respect to the orbital plane. We demonstrate that dust growth is a key factor in the evolution of the H i and H2 mass in disk galaxies under strong RP. We discuss how the correlation between H2 fractions and surface gas densities of disk galaxies evolves with time in the galaxies under RP. We also discuss whether galaxy-wide star formation rates (SFRs) in cluster disk galaxies can be enhanced by RP if the SFRs depend on H2 densities.