Levitation optomechanics exploits the unique mechanical properties of trapped nano-objects in vacuum to address some of the limitations of clamped nanomechanical resonators. In particular, its performance is foreseen to contribute to a better understanding of quantum decoherence at the mesoscopic scale as well as to lead to novel ultrasensitive sensing schemes. While most efforts have focused so far on the optical trapping of low-absorption silica particles, further opportunities arise from levitating objects with internal degrees of freedom, such as color centers. Nevertheless, inefficient heat dissipation at low pressures poses a challenge because most nano-objects, even with low-absorption materials, experience photodamage in an optical trap. Here, by using a Paul trap, we demonstrate levitation in vacuum and center-of-mass feedback cooling of a nanodiamond hosting a single nitrogen-vacancy center. The achieved level of motion control enables us to optically interrogate and characterize the emitter response. The developed platform is applicable to a wide range of other nano-objects and represents a promising step toward coupling internal and external degrees of freedom.