The formation of highly ordered spherical aggregates of silica nanoparticles by the evaporation of single droplets of an aqueous colloidal suspension levitated (confined) in the electrodynamic quadrupole trap is reported. The transient and final structures formed during droplet evaporation have been deposited on a silicon substrate and then studied with SEM. Various successive stages of the evaporation-driven aggregation of nanoparticles have been identified: formation of the surface layer of nanoparticles, formation of the highly ordered spherical structure, collapse of the spherical surface layer leading to the formation of densely packed spherical aggregates, and rearrangement of the aggregate into the final structure of a stable 3D quasi-crystal. The evaporation-driven aggregation of submicrometer particles in spherical symmetry leads to sizes and morphologies of the transient and final structures significantly different than in the case of aggregation on a substrate. The numerical model presented in the article allows us to predict and visualize the observed aggregation stages and their dynamics and the final aggregates observed with SEM.