Multipartite structure, as a critical interaction pattern, is commonly found in diverse real-world networks, in which edges endogenously run only between nodes in different disjoint partites of the network. Disruptive events typically undermine the large-scale connectivity of multipartite networks, and here we show that the self-healing behaviors facilitate efficiently the recovery of the multipartite networks after structural destruction. We develop a theoretical framework to study the robustness of multipartite networks with self-healing capability. After damage, each nonfunctional node will be recovered spontaneously, together with a fraction of the incident edges connected to its intact neighbors in the adjacent partites. We define and compute two key probabilities governing connectivity to the giant component, thereby building our percolation theory of the self-healing multipartite networks with arbitrary degree distributions between adjacent partites and general self-healing ability that is an arbitrary function of the corresponding node degree, and derive the relevant equations that mathematically uncover the intrinsic mechanism of the effects of self-healing behaviors and analytically quantify, for the first time, the robustness of self-healing multipartite networks. The numerical simulation results for a wide range of self-healing schemes and multipartite networks are provided to show the effectiveness and the sharpness of our theoretical results.