Computational multiscale approach, often implemented as the multilevel finite element (FE²) method, provides a rigorous strategy for constructing effective continuum descriptions of heterogeneous composites by explicitly resolving microscale fields within representative volume elements (RVEs). However, when nonlinear mechanisms such as damage and crack evolution or its material nonlinearity are present, FE² method requires recursive micro-macro computations at each macroscopic integration point and load increment, leading to prohibitive costs due to the intricate, history-dependent deformation mechanisms that couple scales. To alleviate this computational burden, we present a multi-task, domain-decomposed physics-informed Deep Operator Network (PI-DeepONet) that learns variational phase-field fracture operators across scales. The fracture process is described by a variational phase-field model in which crack nucleation and propagation arise from minimizing the incremental total energy, which consists of elastic strain energy and fracture dissipation anergy, without any remeshing procedure. Instead of enforcing variational based physics loss, the network is trained to directly minimize the discretized variational energy and boundary terms, thereby aligning the learning objective with finite element implementations. Additionally, transfer learning was employed to carry over physical traits from earlier models, facilitate incremental load steps, and reduce optimization effort. In multiscale coupling, macroscale structural subdomains and RVEs at the microscale are constructed as interconnected tasks within each network for their respective scales. Future work will consider the proposed framework for composite structures under various loading scenarios. This will further demonstrate its potential for efficient multiscale fracture assessment and design of composite structures.