The origin of Cosmic Magnetism is one of the outstanding problems in astrophysics. While galactic dynamos can amplify weak magnetic fields, they require a pre-existing "seed" field (B_seed ~ 10^-20 Gauss) to operate. Standard cosmological models struggle to generate coherent fields on megaparsec scales without invoking exotic inflation scenarios. Unified Field Dynamics (UFD) proposes a mechanical origin based on the global rotation of the universe. We apply the Barnett Effect—the magnetization of an uncharged body by rotation—to the unified vacuum field q. In UFD, the vacuum is a dielectric medium populated by topological charge fluctuations (Sigma dW). We demonstrate that the Global Cosmic Vorticity (Omega), derived in UFD-COS-01 as the source of Dark Energy, induces a Coriolis force on these vacuum charge currents. This breaks the symmetry between positive and negative charge fluctuations, generating a net circular displacement current J_vac. By solving the Maxwell-Coriolis equations for the vacuum, we derive a primordial magnetic field B_seed aligned with the cosmic rotation axis. The predicted magnitude is consistent with the lower bounds required to seed galactic dynamos. Furthermore, this mechanism predicts a Global Helicity in the intergalactic magnetic field, a signature that distinguishes it from random inflationary magnetogenesis.