Fluid-structure interaction (FSI) plays a crucial role in cardiac mechanics, where the strong coupling between fluid flow and deformable structures presents significant computational challenges. The immersed boundary (IB) method efficiently handles large deformations and contact without requiring mesh regeneration. However, solving complex FSI problems demands high computational efficiency, making GPU acceleration essential to leverage massive parallelism, high throughput, and memory bandwidth. We present a fully GPU-accelerated algorithm for the IB method to solve FSI problems in complex cardiac models. The Navier-Stokes equations are discretized using the finite difference method, while the finite element method is employed for structural mechanics. Traditionally, IB methods are not GPU-friendly due to irregular memory access and limited parallelism. The novelty of this work lies in eliminating sparse matrix storage and operations entirely, significantly improving memory access efficiency and fully utilizing GPU computational capability. Additionally, the structural materials can be modeled using general hyperelastic constitutive laws, including fiber-reinforced anisotropic biological tissues such as the Holzapfel-Ogden (HO) model. Furthermore, a combined geometric multigrid solver is used to accelerate the convergence. The full FSI system, consisting of millions of degrees of freedom, achieves a per-timestep computation time of just 0.1 seconds. We conduct FSI simulations of the left ventricle, mitral valve, and aortic valve, achieving results with high consistency. Compared to single-core CPU computations, our fully GPU-accelerated approach delivers speedups exceeding 100 times.