Surgical robot task automation has recently attracted great attention due to its potential to benefit both surgeons and patients. Reinforcement learning (RL) based approaches have demonstrated promising ability to provide solutions to automated surgical manipulations on various tasks. To address the exploration challenge, expert demonstrations can be utilized to enhance the learning efficiency via imitation learning (IL) approaches. However, the successes of such methods normally rely on both states and action labels. Unfortunately action labels can be hard to capture or their manual annotation is prohibitively expensive owing to the requirement for expert knowledge. It therefore remains an appealing and open problem to leverage expert demonstrations composed of pure states in RL. In this work, we present an actor-critic RL framework, termed AC-SSIL, to overcome this challenge of learning with state-only demonstrations collected by following an unknown expert policy. It adopts a self-supervised IL method, dubbed SSIL, to effectively incorporate demonstrated states into RL paradigms by retrieving from demonstrates the nearest neighbours of the query state and utilizing the bootstrapping of actor networks. We showcase through experiments on an open-source surgical simulation platform that our method delivers remarkable improvements over the RL baseline and exhibits comparable performance against action based IL methods, which implies the efficacy and potential of our method for expert demonstration-guided learning scenarios.