Giant molecular clouds (GMCs), where most stars form, may originate from self-gravitating instabilities in the interstellar medium. Using local three-dimensional magnetohydrodynamic simulations, we investigate ways in which galactic turbulence associated with the magnetorotational instability (MRI) may influence the formation and properties of these massive, self-gravitating clouds. Our disk models are vertically stratified with both gaseous and stellar gravity and subject to uniform shear corresponding to a flat rotation curve. Initial magnetic fields are assumed to be weak and purely vertical. For simplicity, we adopt an isothermal equation of state with sound speed cs = 7 km s-1. We find that MRI-driven turbulence develops rapidly, with the saturated-state Shakura & Sunyaev parameter α ~ 0.15-0.3 dominated by Maxwell stresses. Many of the dimensionless characteristics of the turbulence (e.g., the ratio of the Maxwell to Reynolds stresses) are similar to results from previous MRI studies of accretion disks, hence insensitive to the degree of vertical disk compression, shear rate, and the presence of self-gravity—although self-gravity enhances fluctuation amplitudes slightly. The density-weighted velocity dispersions in non- or weakly self-gravitating disks are σx ~ σy ~ (0.4-0.6)cs and σz ~ (0.2-0.3)cs, suggesting that MRI can contribute significantly to the observed level of galactic turbulence. The saturated-state magnetic field strength ~ 2 μG is similar to typical galactic values. When self-gravity is strong enough, MRI-driven high-amplitude density perturbations are swing-amplified to form Jeans-mass (~107 M☉) bound clouds. Compared to previous unmagnetized or strongly magnetized disk models, the threshold for nonlinear instability in the present models occurs for surface densities at least 50% lower, corresponding to the Toomre parameter Qth ~ 1.6. We present evidence that self-gravitating clouds like GMCs formed under conditions similar to our models can lose much of their original spin angular momenta by magnetic braking, preferentially via fields threading nearly perpendicularly to their spin axes. Finally, we discuss the present results within the larger theoretical and observational context, outlining directions for future study.