The implementation of renewable energy sources such as solar photovoltaic (PV), wind, energy storage systems, and fuel cells for electrifying ship transportation systems gives rise to the establishment of an islanded microgrid within the ship. Achieving a balance between demand and generation in this islanded maritime microgrid involves the utilization of a load frequency control (LFC) system, which relies on a communication network for information exchange. The integration of a communication network into shipboard LFC introduces time delays into the feedback loop. Given that renewable energy sources inherently exhibit fluctuating power generation, maintaining frequency stability becomes a critical concern in shipboard microgrid systems. This work focuses on the delay-dependent stability analysis of a shipboard microgrid (MG) system, considering challenges such as unwanted frequency oscillations due to load disturbances and the random nature of renewable resources. In this study, we conduct delay-dependent stability analysis to determine the maximum permissible delay within which the closed-loop system remains asymptotically stable. A less conservative stability criterion is developed using the Lyapunov approach within the linear matrix inequality framework to assess the stability of the closed-loop shipboard LFC system under investigation. To validate the proposed stability criterion, extensive simulations are performed, testing different subsets of the controller parameters against a standard benchmark system. The analytical results are corroborated through comprehensive time-domain simulations, demonstrating the effectiveness of the derived stability criterion in ensuring the stability of the shipboard LFC system under varying conditions.