Will a rod of non-uniform density rotate as it falls?

Gravity acts through the center of mass. Therefore, it cannot exert a torque about the center of mass. Thus, if the rod begins freefall and was not rotating about CoM initially, then it won't rotate about CoM due to gravity.

As different masses fall at the same acceleration of about 9.8 meters per second squared, neglecting air resistance, the lighter half of the rod will be affected by g at the same rate as the heavier half. However non uniform density would imply a greater air resistance to weight ratio from one end to the other. So air resistance, not gravity, could impart a torque to the rod.