To effectively utilize MXenes, a family of two-dimensional materials, in various applications that include thermoelectric devices, semiconductors, and transistors, their thermodynamic and mechanical properties, which are closely related to their stability, must be understood. However, exploring the large chemical space of MXenes and verifying their stability using first-principles calculations are computationally expensive and inefficient. Therefore, this study proposes a machine learning (ML)-based high-throughput MXene screening framework to identify thermodynamically stable MXenes and determine their mechanical properties. A dataset of 23 857 MXenes with various compositions was used to validate this framework, and 48 MXenes were predicted to be stable by ML models in terms of heat of formation and energy above the convex hull. Among them, 45 MXenes were validated using density functional theory calculations, of which 23 MXenes, including Ti2CClBr and Zr2NCl2, have not been previously known for their stability, confirming the effectiveness of this framework. The in-plane stiffness, shear moduli, and Poisson's ratio of the 45 MXenes were observed to vary widely according to their constituent elements, ranging from 90.11 to 198.02 N m-1, 64.00 to 163.40 N m-1, and 0.19 to 0.58, respectively. MXenes with Group-4 transition metals and halogen surface terminations were shown to be both thermodynamically stable and mechanically robust, highlighting the importance of electronegativity difference between constituent elements. Structurally, a smaller volume per atom and minimum bond length were determined to be preferable for obtaining mechanically robust MXenes. The proposed framework, along with an analysis of these two properties of MXenes, demonstrates immense potential for expediting the discovery of stable and robust MXenes.