We consider a repulsive two-component Fermi gas confined in a two-dimensional isotropic harmonic potential and subject to a large Rashba spin-orbit coupling. The single-particle dispersion can be tailored by the spin-orbit-coupling term, which provides an opportunity to study itinerant ferromagnetism in this system. We show that the interplay among spin-orbit coupling, correlation effect, and mean-field repulsion leads to a competition between ferromagnetic and nonmagnetic phases. The weakly correlated nonmagnetic and the ferromagnetic phases can be well described by the mean-field Hartree-Fock theory, while the transition between the ferromagnetic and a strongly correlated nonmagnetic phase is driven by beyond-mean-field quantum correlation effect. Furthermore, the ferromagnetic phase of this system possesses a chiral current density induced by the Rashba spin-orbit coupling, whose experimental signature is investigated.