Direct current microgrids are widely regarded as a promising clean power system technique. However, the microgrid stability is challenged by routine operations and unplanned faults, especially when the operation optimisation overly pursues economy at the expense of stability. This paper proposes a stability-constrained operation optimisation to balance the stability and economy of islanded direct current microgrids. The proposed stability-constrained operation optimisation is formulated as a nonlinear optimisation model and a model-data co-driven solution algorithm is developed. The proposed method strictly guarantees the stability requirements. Furthermore, the proposed method consists of local convex approximations and a small amount of stability evaluation, thus exhibiting high efficiency. Case study illustrates that the proposed method is not affected by prediction errors and avoids the violation of stability constraints. In some tests, the proposed method keeps the solution time within seconds compared to the model-driven method that may take up to the minute level.