While AI-based numerical weather prediction (NWP) enables rapid forecasting, generating high-resolution outputs remains computationally demanding due to limited multi-scale adaptability and inefficient data representations. We propose the 3D Gaussian splatting-based scale-aware vision transformer (GSSA-ViT), a novel framework for arbitrary-resolution forecasting and flexible downscaling of high-dimensional atmospheric fields. Specifically, latitude-longitude grid points are treated as centers of 3D Gaussians. A generative 3D Gaussian prediction scheme is introduced to estimate key parameters, including covariance, attributes, and opacity, for unseen samples, improving generalization and mitigating overfitting. In addition, a scale-aware attention module is designed to capture cross-scale dependencies, enabling the model to effectively integrate information across varying downscaling ratios and support continuous resolution adaptation. To our knowledge, this is the first NWP approach that combines generative 3D Gaussian modeling with scale-aware attention for unified multi-scale prediction. Experiments on ERA5 show that the proposed method accurately forecasts 87 atmospheric variables at arbitrary resolutions, while evaluations on ERA5 and CMIP6 demonstrate its superior performance in downscaling tasks. The proposed framework provides an efficient and scalable solution for high-resolution, multi-scale atmospheric prediction and downscaling. Code is available at: https://github.com/binbin2xs/weather-GS.