We study the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a two-dimensional free-boundary PDE model that generalizes a previous one-dimensional model by combining a Keller-Segel model, Hele-Shaw boundary condition, and the Young-Laplace law with a novel nonlocal regularizing term. This nonlocal term precludes blowup or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We found a family of asymmetric traveling solutions bifurcating from stationary solutions. Our main result is the nonlinear asymptotic stability of traveling wave solutions that model observable steady cell motion. We derived and rigorously justified an explicit asymptotic formula for the stability determining eigenvalue via asymptotic expansions in a small speed of cell. This formula greatly simplifies the computation of this eigenvalue and shows that stability is determined by the change in total myosin mass when stationary solutions bifurcate to traveling solutions. Our spectral analysis reveals the physical mechanisms of stability. If time permits, we will discuss work in progress on fingering instability in multicellular tissue spreading. This is joint work with V. Rybalko and C. Safsten.