Natural selection, the differential reproduction and survival (differential fitness) of individuals due to differences in phenotype, is an essential component of evolution in biological systems. Understanding how natural selection can emerge in non-living systems is critical to understand the emergence of evolution and the origin of life. However, to date, natural selection has only been observed in living systems, or in vitro evolution systems based on templated replication of sophisticated natural biopolymers (nucleic acids). Here we demonstrate that natural selection is possible in a simple chemical system in which competing autocatalytic reactions, compartmentalized in aqueous droplets in an emulsion, are coupled by osmosis and diffusion. A reaction-diffusion model, validated experimentally using the autocatalytic formose reaction, shows that droplets containing efficient autocatalytic reactions can grow, by osmosis, at the expense of droplets containing less efficient autocatalytic reactions. Selective division of droplets that at least double in volume by shearing converts differences in growth rates into differences in fitness. Droplets compete for a common resource (formaldehyde) and growth rate is modulated in an ecological manner, depending on the surrounding droplets. Growth rates before division are correlated with growth rates after division (with formaldehyde re-supplied by nourished with fresh droplets), indicating that the induced variation is heritable. Numerical modelling further indicates that such systems can have multiple attractors (limit cycles), allowing coexistence of different states. Hence, this system displays the basic features of natural selection: two or more entities that grow exponentially at different rates, divide when the volume has doubled, and heritability of growth rates. Our results suggest that natural selection can emerge via simple physicochemical processes and that protocells may not require the chemical sophistication traditionally invoked (e.g., coupling of RNA self-replication to protocell replication). They may also pave the way to implementing evolutionary mechanisms in synthetic chemical systems.