In this paper, we introduce a novel and robust approach to quantized matrix completion. First, we propose a rank minimization problem with constraints induced by quantization bounds. Next, we form an unconstrained optimization problem by regularizing the rank function with Huber loss. Huber loss is leveraged to control the violation from quantization bounds due to two properties: first, it is differentiable; and second, it is less sensitive to outliers than the quadratic loss. A smooth rank approximation is utilized to endorse lower rank on the genuine data matrix. Thus, an unconstrained optimization problem with differentiable objective function is obtained allowing us to advantage from gradient descent technique. Novel and firm theoretical analysis of the problem model and convergence of our algorithm to the global solution are provided. Another contribution of this letter is that our method does not require projections or initial rank estimation, unlike the state-of-the-art. In the Numerical Experiments section, the noticeable outperformance of our proposed method in learning accuracy and computational complexity compared to those of the state-of-the-art literature methods is illustrated as the main contribution.