Identify the force that caused stars and planets in the universe to become layered according to density differences in their compositions

The answer is gravity. According to general relativity by Einstein, the bending of space-time is dependent on the nature of the matter of the object, or energy, momentum, and radiation. The heavier the object, the bigger the curvature of space-time it causes, that is felt as gravity. Heavier objects with a bigger gravity pull, pull other objects (with lesser gravity pull) more towards them than vice versa.