Isolated hydrogen atoms absorbed on graphene are predicted to induce magnetic moments. Here we demonstrate that the adsorption of a single hydrogen atom on graphene induces a magnetic moment characterized by a similar to 20-millielectron volt spin-split state at the Fermi energy. Our scanning tunneling microscopy (STM) experiments, complemented by first-principles calculations, show that such a spin-polarized state is essentially localized on the carbon sublattice opposite to the one where the hydrogen atom is chemisorbed. This atomically modulated spin texture, which extends several nanometers away from the hydrogen atom, drives the direct coupling between the magnetic moments at unusually long distances. By using the STM tip to manipulate hydrogen atoms with atomic precision, it is possible to tailor the magnetism of selected graphene regions.