Designing multiscale structures entails several challenging tasks, including generating random material microstructures, developing microstructure-resolved physics models, and performing multiscale design and optimization. Conventionally, multiscale finite element (FE) methods are implemented to perform microstructure-resolved physics simulations. However, these slow and computationally expensive numerical methods often become a bottleneck in multiscale design and uncertainty quantification of complex material systems. To overcome these computational challenges, we present an integrated framework combining deep learning (DL) and numerical optimization for accelerated multiscale modeling and design of material with enhanced performance. The proposed framework utilizes a 3D U-Net convolutional neural network to map the microstructures to the corresponding physical fields. This DL framework is integrated with a numerical optimization approach to design materials with targeted properties. Through several numerical examples, we showcase the applicability of this DL framework to design copper alloy microstructures with improved spall strength and validate the results by conducting demanding laser-driven spall experiments on thin metal foils. Our work enhances decision-making in material design and testing by combining the speed of AI with high-throughput experiments, thereby accelerating the design of optimal materials.