We consider the problem of finding a singularity of a vector field $X$ on a complete Riemannian manifold. In this regard we prove a unified result for local convergence of Newton's method. Inspired by previous work of Zabrejko and Nguen on Kantorovich's majorant method, our approach relies on the introduction of an abstract one-dimensional Newton's method obtained using an adequate Lipschitz-type radial function of the covariant derivative of $X$. The main theorem gives in particular a synthetic view of several famous results, namely the Kantorovich, Smale and Nesterov-Nemirovskii theorems. Concerning real-analytic vector fields an application of the central result leads to improvements of some recent developments in this area.