Accurate 3D aortic construction is crucial for clinical diagnosis, preoperative planning, and computational fluid dynamics (CFD) simulations, as it enables the estimation of critical hemodynamic parameters such as blood flow velocity, pressure distribution, and wall shear stress. Existing construction methods often rely on large annotated training datasets and extensive manual intervention. While the resulting meshes can serve for visualization purposes, they struggle to produce geometrically consistent, well-constructed surfaces suitable for downstream CFD analysis. To address these challenges, we introduce AortaDiff, a diffusion-based framework that generates smooth aortic surfaces directly from CT/MRI volumes. AortaDiff first employs a volume-guided conditional diffusion model (CDM) to iteratively generate aortic centerlines conditioned on volumetric medical images. Each centerline point is then automatically used as a prompt to extract the corresponding vessel contour, ensuring accurate boundary delineation. Finally, the extracted contours are fitted into a smooth 3D surface, yielding a continuous, CFD-compatible mesh representation. AortaDiff offers distinct advantages over existing methods, including an end-to-end workflow, minimal dependency on large labeled datasets, and the ability to generate CFD-compatible aorta meshes with high geometric fidelity. Experimental results demonstrate that AortaDiff performs effectively even with limited training data, successfully constructing both normal and pathologically altered aorta meshes, including cases with aneurysms or coarctation. This capability enables the generation of high-quality visualizations and positions AortaDiff as a practical solution for cardiovascular research.