We simulate liquid-vapor phase separation in a van der Waals fluid that is deeply quenched into the unstable range of its phase diagram. Our theoretical approach follows the diffuse-interface model, where convection induced by phase change is accounted for via a non-equilibrium (Korteweg) force expressing the tendency of the liquid-vapor system to minimize its free energy. Spinodal decomposition patterns for critical and off-critical van der Waals fluids are studied numerically, revealing the scaling laws of the characteristic lengthscale and composition of single-phase microdomains, together with their dependence on the Reynolds number. Unlike phase separation of viscous binary mixtures, here local equilibrium is reached almost immediately after single-phase domains start to form. In addition, as predicted by scaling laws, such domains grow in time like t^2/3. Comparison between 2D and 3D results reveals that 2D simulations capture, even quantitatively, the main features of the phenomenon.