Deterministic lateral displacement (DLD) is a passive separation method that separates particles by hydrodynamic size. This label-free method is a promising technique for cell separation because of its high size resolution and insensitivity to flow rate. Development of capillary-driven microfluidic technologies allows microfluidic devices to be operated without any external power for fluid pumping, lowering their total cost and complexity. Herein, we develop and test a DLD-based particle and cell sorting method that is driven entirely by capillary pressure. We show microchip self-filling, flow focusing, flow stability, and capture of separated particles. We achieve separation efficiency of 92% for particle-particle separation and more than 99% efficiency for cell-particle separation. The high performance of driven flow and separation along with simplicity of the operation and setup make it a valuable candidate for point-of-care devices.