Denoising diffusion probabilistic models (DDPMs) have pioneered new state-of-the-art results in disciplines such as computer vision and computational biology for diverse tasks ranging from text-guided image generation to structure-guided protein design. Along this latter line of research, methods have recently been proposed for generating 3D molecules using equivariant graph neural networks (GNNs) within a DDPM framework. However, such methods are unable to learn important geometric properties of 3D molecules, as they adopt molecule-agnostic and non-geometric GNNs as their 3D graph denoising networks, which notably hinders their ability to generate valid large 3D molecules. In this work, we address these gaps by introducing the Geometry-Complete Diffusion Model (GCDM) for 3D molecule generation, which outperforms existing 3D molecular diffusion models by significant margins across conditional and unconditional settings for the QM9 dataset and the larger GEOM-Drugs dataset, respectively, and generates more novel and unique unconditional 3D molecules for the QM9 dataset compared to previous methods. Importantly, we demonstrate that the geometry-complete denoising process of GCDM learned for 3D molecule generation enables the model to generate a significant proportion of valid and energetically-stable large molecules at the scale of GEOM-Drugs, whereas previous methods fail to do so with the features they learn. Additionally, we show that extensions of GCDM can not only effectively design 3D molecules for specific protein pockets but also that GCDM's geometric features can be repurposed to consistently optimize the geometry and chemical composition of existing 3D molecules for molecular stability and property specificity, demonstrating new versatility of molecular diffusion models. Our source code and data are freely available at https://github.com/BioinfoMachineLearning/Bio-Diffusion.