Contact characteristic is an important index to characterize the transmission performance of gears. To clarify the contact characteristic of a nutation transmission with double circular-arc spiral bevel gears, this paper carried out a loaded tooth contact analysis by using the finite element method. Firstly, based on the meshing theory, the tooth surface equation of the double circular-arc spiral bevel gear was derived based on which a three-dimensional CAD model of the meshing gear pair was generated. Secondly, a finite element model of the gear pair was developed in Ansys Workbench to simulate contact status of the meshing double circular-arc spiral bevel gear pair. Finally, the influence of load and installation error on the transmission performance of the gear pair was investigated. The results show that a double-point contact occurs on concave and convex tooth surfaces of the double circular-arc spiral bevel gear, and the stress maximum occurs at the toe of bevel gear. With the increase of load, the contact ratio of the meshing gear increases, while the load distribution coefficient decreases, which improves the smoothness of gear transmission. The installation error affects the contact state of the gear pair in that a negative offset error and a positive nutation angle error may degrade the bearing capacity and the transmission stability of the gears. Thus, they should be reasonably controlled in practical applications.