Phase I clinical studies are experiments in which a new drug is administered to humans to determine the maximum dose that causes toxicity with a target probability. Phase I dose-finding is often formulated as a quantile estimation problem. For studies with a biological endpoint, it is common to define toxicity by dichotomizing the continuous biomarker expression. In this article, we propose a novel variant of the Robbins-Monro stochastic approximation that utilizes the continuous measurements for quantile estimation. The Robbins-Monro method has seldom seen clinical applications, because it does not perform well for quantile estimation with binary data and it works with a continuum of doses that are generally not available in practice. To address these issues, we formulate the dose-finding problem as root-finding for the mean of a continuous variable, for which the stochastic approximation procedure is efficient. To accommodate the use of discrete doses, we introduce the idea of virtual observation that is defined on a continuous dosage range. Our proposed method inherits the convergence properties of the stochastic approximation algorithm and its computational simplicity. Simulations based on real trial data show that our proposed method improves accuracy compared with the continual re-assessment method and produces results robust to model misspecification.