Spin-mechanical hybrid systems have been widely used in quantum information processing. However, the spin-mechanical interaction is generally weak, making it a critical challenge to enhance the spin-mechanical interaction into the strong coupling or even ultrastrong coupling regime. Here, we propose a protocol that can significantly enhance the spin-mechanical coupling strength with a diamond spin vacancy and a levitated micromagnet. A driving electrical current is used to modulate the mechanical motion of the levitated micromagnet, which induces a two-phonon drive and can exponentially enhance the spin-phonon and phonon-medicated spin-spin coupling strengths. Furthermore, a high-fidelity Schrödinger cat state and an unconventional two-qubit geometric phase gate with high fidelity and faster gate speed can be achieved using this hybrid system. This protocol provides a promising platform for quantum information processing with nitrogen-vacancy spins coupled to levitated micromagnets.