The hierarchical structures found in biological materials lead to an outstanding balance of multiple material properties, and numerous research studies have been initiated to emulate the key concepts for the designing of engineering materials, the so-called bioinspired composites. However, the optimization of bioinspired composites has long been difficult as it usually falls into the category of 'black-box problem', the objective functions not being available in a functional form. Also, bioinspired composites possess multiple material properties that are in a trade-off relationship, making it impossible to reach a unique optimal design solution. As a breakthrough, we propose a data-driven material design framework which can generate bioinspired composite designs with an optimal balance of material properties. In this study, a nacre-inspired composite is chosen as the subject of study and the optimization framework is applied to determine the designs that have an optimal balance of strength, toughness, and specific volume. Gaussian process regression was adopted for the modeling of a complex input-output relationship, and the model was trained with the data generated from the crack phase-field simulation. Then, multi-objective Bayesian optimization was carried out to determine pareto-optimal composite designs. As a result, the proposed data-driven algorithm could generate a 3D pareto surface of optimal composite design solutions, from which a user can choose a design that suits his/her requirement. To validate the result, several pareto-optimal designs are built using a PolyJet 3D printer, and their tensile test results show that each of the characteristic designs is well optimized for its specific target objective.