As the penetration rate of renewable energy generation continues to increase, the characteristics of weak grids with low short-circuit ratios (SCR) and high grid impedance at the distribution end become more pronounced. Although the performance of grid-connected inverters can be adaptively adjusted based on SCR to ensure stable operation over a wide range of SCR variations, this significantly reduces their responsiveness, making it difficult to meet grid connection requirements. To address this issue, this paper proposes a control strategy based on a fuzzy adaptive PI controller. First, a mathematical model of the grid-connected inverter is established, and the D-partition method is used to analyze the stability region of conventional PI control parameters, particularly the balance between stability and responsiveness in weak grids. On this basis, a fuzzy adaptive PI controller that combines SCR and current control error is designed, capable of autonomously adjusting control parameters according to the steady-state and dynamic characteristics of the grid. Finally, the effectiveness of this strategy is verified through a hardware-in-the-loop (HIL) simulation platform. The results show that the proposed method improves the stability and dynamic response capability of grid-connected inverters under weak grid conditions.