Phase-field models for brittle fracture provide high-fidelity predictions of complex crack propagation but remain computationally prohibitive for large-scale engineering applications. This work introduces a novel hybrid framework that bridges physics-based simulation with data-driven methodologies, coupling the FEM solver Akantu with a Graph-based Integral Neural Operator (GINO). Our approach implements a predictor-corrector scheme wherein the GINO, trained on high-fidelity simulation data, predicts the material damage field at subsequent time steps. This prediction serves as an accurate initial guess for Newton-Raphson refinement within the FEM solver, ensuring physical consistency while potentially reducing computational cost. This tight coupling demonstrates a practical pathway for integrating deep learning models with established physics solvers. The neural operator achieves strong predictive accuracy, with a mean error of 0.4% on the damage field across validation scenarios, confirming its ability to learn the complex, non-linear dynamics of crack evolution. Our optimized implementation leverages efficient graph-based operations to handle irregular finite element meshes. Evaluation on NVIDIA A100 GPU provides crucial proof-of-concept that neural operators can effectively learn and predict phase-field physics in a coupled environment. The primary contribution is the successful integration of a physics-informed neural operator with a production FEM solver for crack propagation. This proof-of-concept demonstrates that ML-accelerated approaches can achieve computational efficiency while maintaining physical fidelity, establishing a pathway for future developments in computational fracture mechanics.