We demonstrated the operation of an acoustomicrofluidic device composed of a polydimethylsiloxane (PDMS) microchannel and a slanted-finger interdigitated transducer (SF-IDT), for the on-demand splitting of droplets in an active, accurate, rapid, and size-controllable manner. A narrow beam of surface acoustic waves (SAWs) that emanated from the SF-IDT exerted an acoustic radiation force (ARF) on the droplet's water-oil interface due to the acoustic contrast between the two fluids. The ARF split the mother droplet into two or more daughter droplets of various volumes in a split ratio that was readily controlled by varying the applied voltage or the flow rate. Theoretical estimates of the ARF acting on the droplet interface were used to investigate the mechanism underlying the droplet splitting properties and size control. The versatility of the acoustomicrofluidic device operation was demonstrated by selectively pushing/placing a suspended polystyrene particle into a specific/preferred split daughter droplet using the direct ARF acting on the particle.