Inverse microstructure design plays a central role in materials discovery, yet remains challenging due to the complexity of structure–property linkages and the scarcity of labeled training data. We propose Design-GenNO, a physics-informed generative neural operator framework that unifies generative modeling with operator learning to address these challenges. Microstructures are encoded into a low-dimensional, structured latent space, which serves as a generator for both reconstructing microstructures and predicting PDE solution fields. MultiONet-based decoders enable functional mappings from latent variables to both microstructures and full PDE solution fields, thereby supporting a wide range of inverse design objectives without retraining. A normalizing flow prior regularizes the latent space, facilitating efficient sampling and gradient-based optimization. A distinctive feature of the framework is its physics-informed training strategy, in which PDE residuals are embedded directly into the learning objective, enabling training with limited or even no labeled field data. We validate the method on inverse design tasks in two-phase materials, including effective property matching, recovery of microstructures from sparse field measurements, and maximization of conductivity ratios. Across all tasks, Design-GenNO achieves high accuracy, generates diverse and physically meaningful designs, and consistently outperforms state-of-the-art methods. Moreover, it demonstrates strong extrapolative capabilities by producing microstructures with effective properties beyond those present in the training distribution. These results establish Design-GenNO as a robust and general framework for physics-informed inverse design, offering a promising pathway toward accelerated materials discovery.