The inverse design of spinodoid architected materials — characterized by smooth, non-periodic, and morphologically complex microstructures — remains a critical challenge due to the high dimensionality of the morphological parameter space and the prohibitive cost of high-fidelity numerical models. To address the challenges associated with the inverse design of the spinodoid structures, an uncertainty-aware Bayesian framework is developed, integrating Bayesian optimization and Bayesian inversion within a unified, surrogate-assisted strategy. Bayesian optimization based on Gaussian process surrogate modeling is employed to adaptively explore the structure–property landscape using a limited number of high-fidelity model evaluations, enabling data-efficient and accurate approximation of the underlying response. The resulting surrogate model is subsequently embedded into a Bayesian inversion formulation to infer the posterior distribution of morphological parameters conditioned on prescribed target properties, with uniform priors adopted to reflect epistemic uncertainty in the absence of prior information. The inverse design is reformulated as a data-consistent least-squares minimization over the surrogate model predictions, yielding statistically calibrated estimates of optimal designs together with quantified predictive uncertainties. The effectiveness and scalability of the framework are demonstrated through applications to inverse design problems for spinodoid structures in bi-dimensional and four-dimensional parameter spaces, enabling the identification of structures with tailored mechanical functionalities. The results confirm the capability of the proposed methodology to deliver accurate, interpretable, and computationally tractable characterization of optimal or most probable spinodoid configurations, thereby advancing the uncertainty-aware design of architected materials.