We introduce a matrix-free numerical continuation framework for the Ikeda map of coherently driven passive Kerr resonators. The framework traces solution branches and evaluates their linear stability without explicitly assembling the full Jacobian matrices, using a unified formulation for states that recur after one or multiple round trips. We resolve stable and unstable branches of modulation-instability patterns and temporal cavity solitons, and uncover period-doubled homogeneous states and associated multi-round-trip dynamics that are absent from the standard Lugiato-Lefever equation (LLE). For period-1 states, the Ikeda map and the LLE show close agreement in the high-finesse regime, while their branch-existence intervals increasingly deviate as the finesse decreases. These results demonstrate that direct continuation of the unaveraged round-trip map can capture nonlinear states and multi-round-trip dynamics beyond the mean-field description, providing a systematic framework for bifurcation and stability analysis of Kerr resonators without relying on the averaged LLE.