Many economic models feature monotone Markov dynamics on state spaces that may be noncompact. Establishing existence, uniqueness, and stability of stationary distributions in such settings has required a patchwork of sufficient conditions, each tailored to specific applications. We provide a single necessary and sufficient condition: a monotone Markov process has a globally stable stationary distribution if and only if it is asymptotically contractive and has a tight trajectory. This characterization covers both compact and noncompact state spaces, discrete and continuous time, and extends to nonlinear Markov operators that depend on aggregate state. We demonstrate the result through applications to wage dynamics, Bayesian learning with belief shocks, and income processes that generate Pareto tails.