The effects of minimum stress on cumulative damage and fatigue lifetime were assessed for an angle-plied nylon-cord-reinforced elastomer composite that represents the bias aircraft tire carcass. The S-N curves were established under constant minimum stress rather than constant R-ratio. The trends of longer fatigue life at a given stress range and higher fatigue endurance limit of the composite were clearly observed with a higher level of minimum stress. These trends stem from the fact that the stress and strain are not linearly related to each other. The stress-strain curve of the elastomer matrix composite exhibits strain stiffening. As a result, in a stress-controlled fatigue test, a shorter strain range or a smaller amount of strain energy is experienced for the same stress range when the level of minimum stress is raised. However, it should be noted that the levels of minimum stress used in the study happened to be safely below a critical level for the initiation of fiber-matrix debonding under static tension. As in other structural materials, the increase of applied stress level in elastomers can be regarded as an increase of the damage potential. Therefore, when the minimum stress exceeds a critical level, the damage potential effect may become dominant and a shorter fatigue life at a given stress range could be observed with a higher level of minimum stress in S-N curves. In the stress-controlled fatigue tests, the role of damage potential could be assessed more clearly when the fatigue life data were plotted directly against the damage parameters, such as strain range, dynamic creep rate, dissipated strain energy, and stored strain energy. The fatigue life curves based on the damage parameters at midlife showed a clear trend of a shorter fatigue life with a higher level of minimum stress. This trend could be established at a given strain range, dynamic creep rate, or dissipated strain energy. The results are in striking contrast to the trend of the S-N curves. The trend of a shorter fatigue life with a higher level of minimum stress was less obvious at a given stored strain energy. The values of dynamic creep rate, dissipated strain energy, and stored strain energy at midlife were found to have a power law relationship with the fatigue life of aircraft tire carcass composite.