The ionosphere is a vitally dynamic charged particle region in the Earth's upper atmosphere, playing a crucial role in applications such as radio communication and satellite navigation. The slant total electron contents (STECs) are an important parameter for characterizing wave propagation, representing the integrated electron density along the ray of radio signals passing through the ionosphere. The accurate prediction of STEC is essential for mitigating the ionospheric impact particularly on Global Navigation Satellite Systems (GNSS). In this work, we propose a high-precision STEC prediction model named deep neural operator network (DeepONet)-STEC, which learns nonlinear operators to predict the 4-D temporal-spatial integrated parameter for the specified satellite-ground station ray path globally. As a demonstration, we validate the performance of the model based on GNSS observation data for global and US Continuously Operating Reference Stations (CORS) regimes under ionospheric quiet and storm conditions. The DeepONet-STEC model results show that the three-day 72 h prediction in quiet periods could achieve high accuracy using observation data by the precise point positioning (PPP) with temporal resolution <inline-formula> <tex-math notation="LaTeX">30~\rm {s} </tex-math></inline-formula>. Under active solar magnetic storm periods, the DeepONet-STEC also demonstrated its robustness and superiority than traditional deep learning methods. This work presents a neural operator regression architecture for predicting the 4-D spatiotemporal ionospheric state for satellite navigation system performance, which may be further extended for various space applications and beyond.