The power grid and natural gas network can achieve highly reliable operation through bidirectional coupling. To solve the fault self-healing problem of the electrical-gas coupled energy internet, a self-healing and optimized operation method of the energy internet is proposed. First, based on the electrical-gas coupling characteristics, the method makes full use of the natural gas network to supplement the grid energy, and considers the economy of the natural gas network and the uncertainty of new energy output to realize fault fast self-healing and optimizing operation of the integrated energy system. The upper model uses the improved ant colony algorithm based on the breadth-first search method to optimize the power supply recovery path to obtain the system switching state. The lower model is based on the analysis of the electrical-gas coupling characteristics and takes the economics of the natural gas network as the main goal. Then it uses the conditional value-at-risk theory (CVaR), and at the same time considers the operational risk brought by the uncertainty of new energy output to build an optimization and reconstruction model of the electricity-gas interconnected energy internet. Then, the two-layer optimization model is analysed and global optimization is carried out. Then optimal fault recovery and operation of the electricity-gas interconnected energy dispatch scheme is obtained. Finally, the effectiveness of the proposed method is verified by an energy internet simulation model coupled with an IEEE 33-node distribution network and a 7-node natural gas network. This work is supported by the National Natural Science Foundation of China (No. 61873159).