Monocular depth estimation (MDE) has been widely adopted in the perception systems of autonomous vehicles and mobile robots. However, existing approaches often struggle to maintain temporal consistency in depth estimation across consecutive frames. This inconsistency not only causes jitter but can also lead to estimation failures when the depth range changes abruptly. To address these challenges, this paper proposes a consistency-aware monocular depth estimation framework that leverages wheel odometry from a mobile robot to achieve stable and coherent depth predictions over time. Specifically, we estimate camera pose and sparse depth from triangulation using optical flow between consecutive frames. The sparse depth estimates are used to update a recursive Bayesian estimate of the metric scale, which is then applied to rescale the relative depth predicted by a pre-trained depth estimation foundation model. The proposed method is evaluated on the KITTI, TartanAir, MS2, and our own dataset, demonstrating robust and accurate depth estimation performance.