We present a novel fully coupled monolithic solver for the direct-forcing immersed boundary method (IBM) for incompressible flows. The solver targets the standard saddle-point system. It treats pressure, velocity, nonlinear convection terms, and Lagrangian forces within a single framework by utilizing a modified big-box Vanka smoother. This smoother is extended with additional Lagrange multipliers arising from the IBM formulation. A key component enabling the enhanced efficiency of the developed methodology is the use of a Schur-complement decomposition. It reduces the operator size by two-thirds while preserving stability and accuracy. The monolithic formulation eliminates splitting errors. It also avoids artificial pressure boundary conditions, which are common in segregated methods. The approach supports stable operation at high CFL numbers (up to 0.5). This makes it well suited for moving-boundary simulations. We verify the solver on a broad set of benchmark problems. The tests include stationary and moving immersed bodies. They cover a wide range of Reynolds numbers. The results show computational times comparable to existing semi-implicit methods. At the same time, accuracy and stability are improved. The proposed method therefore provides a robust and broadly applicable monolithic solver for incompressible IBM simulations.