Using extensive kinematical and morphological data on two Virgo Cluster galaxies undergoing strong nuclear star formation, we show that ram pressure stripping and gravitational interactions can act together on galaxies that have recently fallen into clusters. We present a detailed study of the peculiar H I-deficient Virgo Cluster spiral galaxies NGC 4064 and NGC 4424 using ¹²CO 1-0 interferometry, optical imaging, and integral field spectroscopic observations in order to learn what type of environmental interactions have affected these galaxies. Optical imaging reveals that NGC 4424 has a strongly disturbed stellar disk, with banana-shaped isophotes and shells. NGC 4064, which lies in the cluster outskirts, possesses a relatively undisturbed outer stellar disk and a central bar. In both galaxies Hα emission is confined to the central kiloparsec and originates in barlike strings of luminous star-forming complexes surrounded by fainter filaments. Complexes of young blue stars exist beyond the present location of ongoing star formation, indicating rapidly shrinking star-forming disks. Disturbed dust lanes extend out to a radius of 2-3 kpc, much farther than the Hα and CO emission detected by us but similar to the blue stellar complexes. CO observations reveal bilobal molecular gas morphologies, with Hα emission peaking inside the CO lobes, implying a time sequence in the star formation process. Gas kinematics reveals strong barlike noncircular motions in the molecular gas in both galaxies, suggesting that the material is radially infalling. In NGC 4064 the stellar kinematics reveals strong barlike noncircular motions in the central 1 kpc and stars supported by rotation with V/σ > 1 beyond a radius of 15'' (1.2 kpc). On the other hand, NGC 4424 has extremely modest stellar rotation velocities (V_max ~ 30 km s^-1), and stars are supported by random motions as far out as we can measure, with V/σ = 0.6 at r = 18'' (1.4 kpc). The ionized gas kinematics in the core are disturbed and possibly counterrotating. The observations suggest that the peculiarities of NGC 4424 are the result of an intermediate-mass merger plus ram pressure stripping. In the case of NGC 4064, the evidence suggests an already stripped "truncated/normal" galaxy that recently suffered a minor merger or tidal interaction with another galaxy. Observations of the present star formation rate and gas content suggest that these galaxies will become small-bulge S0s within the next 3 Gyr. We propose that galaxies with "truncated/compact" Hα morphologies such as these are the result of the independent effects of ram pressure stripping, which removes gas from the outer disk, and gravitational interactions such as mergers, which heat stellar disks, drive gas to the central kiloparsec, and increase the central mass concentrations. Together these effects transform the morphology of these galaxies.