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As more practical and scalable quantum computers emerge, much attention has been focused on realizing quantum supremacy in machine learning. Existing quantum ML methods either (1) embed a classical model into a target Hamiltonian to enable quantum optimization or (2) represent a quantum model using variational quantum circuits and apply classical gradient-based optimization. The former method leverages the power of quantum optimization but only supports simple ML models, while the latter provides flexibility in model design but relies on gradient calculation, resulting in barren plateau (i.e., gradient vanishing) and frequent classical-quantum interactions. To address the limitations of existing quantum ML methods, we introduce Quark, a gradient-free quantum learning framework that optimizes quantum ML models using quantum optimization. Quark does not rely on gradient computation and therefore avoids barren plateau and frequent classical-quantum interactions. In addition, Quark can support more general ML models than prior quantum ML methods and achieves a dataset-size-independent optimization complexity. Theoretically, we prove that Quark can outperform classical gradient-based methods by reducing model query complexity for highly non-convex problems; empirically, evaluations on the Edge Detection and Tiny-MNIST tasks show that Quark can support complex ML models and significantly reduce the number of measurements needed for discovering near-optimal weights for these tasks.