In a 100% renewable energy scenario, islanded microgrids lack the support of conventional generation, leading to more severe frequency and voltage fluctuations, while small-signal stability also becomes a critical challenge. To address this issue in inverter-dominated renewable islanded microgrids, an optimal operation method incorporating small-signal stability constraints based on an input convex neural network (ICNN) is proposed. First, an optimal scheduling model is established considering both inverter P-f and Q-V droop control. Information-gap decision theory (IGDT) is employed to handle uncertainties in generation and load. Then, using damping ratio sensitivity as the direction for operating point updates, sampling points are iteratively shifted to efficiently approach the stability boundary region, thereby obtaining a large number of stability boundary samples under simultaneous variations of droop control parameters. Next, the ICNN is used to learn the mapping between droop control parameters and small-signal stability indices. The nonlinear stability constraints are subsequently linearized and embedded into the scheduling model. Finally, case studies on the IEEE 33-bus islanded microgrid test system demonstrate the feasibility and accuracy of the proposed method.