While mixed-integer linear programming and convex programming solvers have advanced significantly over the past several decades, solution technologies for general mixed-integer nonlinear programs (MINLPs) have yet to reach the same level of maturity. Various problem structures across different application domains remain challenging to model and solve using modern global solvers, primarily due to the lack of efficient parsers and convexification routines for their complex algebraic representations. In this paper, we introduce a novel graphical framework for globally solving MINLPs based on decision diagrams (DDs), which enable the modeling of complex problem structures that are intractable for conventional solution techniques. We describe the core components of this framework, including a graphical reformulation of MINLP constraints, convexification techniques derived from the constructed graphs, efficient cutting plane methods to generate linear outer approximations, and a spatial branch-and-bound scheme with convergence guarantees. In addition to providing a global solution method for tackling challenging MINLPs, our framework addresses a longstanding gap in the DD literature by developing a general-purpose DD-based approach for solving general MINLPs. To demonstrate its capabilities, we apply our framework to solve instances from one of the most difficult classes of unsolved test problems in the MINLP Library, which are otherwise inadmissible for state-of-the-art global solvers.