In this paper, a methodology to analyze the nonlinear vibration of a Jeffcott rotor supported on a squeeze-film damper (SFD) with centering springs, which is widely used in high-speed rotating machines, is proposed using the incremental harmonic balance (IHB) method. In this paper, the IHB method is modified to analyze the dynamic behavior of rotor systems supported on fluid-film bearings, such as SFDs. The processing of the mass matrix, stiffness matrix, and linear force matrix proceeds in exactly the same way as in classical IHB. The nonlinear force generated in the oil film of the SFD is calculated using the alternating frequency/time method and the transformation matrix and incorporated into the computational processing of the classical IHB method. This calculation method is first proposed in this paper. Solutions computed using the proposed method are compared with solutions computed using numerical integration. The results are very close. The stability of the calculated solutions is determined using the Floquet theory. Based on this, frequency–response curves according to the change in various parameters are constructed. The proposed method can be effectively used to analyze the nonlinear vibration characteristics of rotor systems supported on fluid-film bearings, such as SFDs.