Rotating machinery is inherently susceptible to costly and dangerous faults. One such commonly encountered fault is undesirable dynamic contact between the rotor and stator (i.e., rotor–stator rub). The forces generated during rotor–stator rub are fundamentally tribological, as they are generated by contact and friction and result in wear. These forces are typically found by assuming linear elastic contact and dry Coulomb friction at the rotor–stator interface, where the normal force is a linear function of the interference. For the first time, this work incorporates viscoelasticity into the stator support and investigates its influence on the global dynamics of rotor–stator rub. The viscoelastic stator supports are modeled using fractional calculus, an approach which adeptly and robustly characterizes the viscoelasticity. Specifically, a fractional derivative order of one-half is employed to generate an analytic time-domain form of viscoelastic impedance. This approach directly assimilates viscoelasticity into the system dynamics, since the rotor equations of motion are integrated numerically in the time-domain. The coupled rotor–stator dynamic model incorporating viscoelastic supports is solved numerically to explore the influence of viscoelasticity. This model provides a framework for analysis of dynamic systems where viscoelasticity is included.