Microscale flow plays an important role in several areas, including microbiological systems and microfluidic devices. These systems are often placed in viscous or complex fluids such as polymer solutions. Understanding microscale flow in viscous fluids will lead to a further development of microfluidic devices and elucidation of the collective motion of microorganisms. We studied the microscale flow induced by the optically driven rotation of a nematic liquid crystal (NLC) droplet in an aqueous glycerol solution. The rotation of the droplets was controlled using circularly polarized optical tweezers. In water, the induced flow agrees well with the theoretical flow assuming a solid rotating particle and a no-slip boundary condition. However, the induced flow velocity deviated from the theoretical value as the viscosity of the glycerol solution increased. This deviation was mainly due to slip on the droplet surface. As an application of the NLC rotator, the viscosity of the solutions and the hydrodynamic interactions between the two rotating particles were measured.