We study the wetting behavior on spherical walls by ternary mixtures of oil, water, and an amphiphile. We use the Ginzburg-Landau free energy with a single order parameter and find that there are different stable structures of the interface and that a quasiwetting transition is the mechanism involved in the transition among them. We calculate these wetting transitions for two sets of parameters in the bulk free energy which are known to show microemulsion behavior. The surface transitions are thin-thick first-order transitions (continuous transitions are absent), and the phase diagram in surface parameter space is constructed. For the first set of bulk parameters water, oil, and a microemulsion coexist, and we study the first-order transition where the oil phase wets the wall-microemulsion interface and its behavior as the radius of the wall becomes large. Therefore, we recover the known wetting transitions on a planar wall. In the second set of bulk parameters only water and oil coexist, and for some sizes of the solid wall, the oil phase wets the wall-water interface, and the phase behavior is extremely rich. We obtain a coexistence of four surface phases or two triple points followed by three lines of first-order transitions which end at three critical points depending on the radius of the surface. When there are micellar metastable solutions in bulk, the behavior of the thickness of the wetting layer of the oil phase as the radius of the spherical wall gets larger is nonmonotonic. We associate this behavior with the intrinsic micelle structure due to the spontaneous curvature of the model.