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We present a new method for calculation of the gravitational self-force (GSF) in Kerr geometry, based on a time-domain reconstruction of the metric perturbation from curvature scalars. In this approach, the GSF is computed directly from a certain scalar-like self-potential that satisfies the time-domain Teukolsky equation on the Kerr background. The approach is computationally much cheaper than existing time-domain methods, which rely on a direct integration of the linearized Einstein's equations and are impaired by mode instabilities. At the same time, it retains the utility and flexibility of a time-domain treatment, allowing calculations for any type of orbit (including highly eccentric or unbound ones) and the possibility of self-consistently evolving the orbit under the effect of the GSF. Here we formulate our method, and present a first numerical application, for circular geodesic orbits in Schwarzschild geometry. We discuss further applications.