Since humans and robots are increasingly sharing portions of their operational spaces, experimental evidence is needed to ascertain the safety and social acceptability of robots in human-populated environments. Although several studies have aimed at devising strategies for robot trajectory planning to perform safe motion in populated environments, a few efforts have measured to what extent a robot trajectory is accepted by humans. Here, we present a navigation system for autonomous robots that ensures safety and social acceptability of robotic trajectories. We overcome the typical reactive nature of state-of-the-art trajectory planners by leveraging non-cooperative game theory to design a planner that encapsulates human-like features of preservation of a personal space, recognition of groups, sequential and strategized decision making, and smooth obstacle avoidance. Social acceptability is measured through a variation of the Turing test administered in the form of a survey questionnaire to a pool of 691 participants. Comparison terms for our tests are a state-of-the-art navigation algorithm (Enhanced Vector Field Histogram, VFH) and purely human trajectories. While all participants easily recognized the non-human nature of VFH-generated trajectories, the distinction between game-theoretical trajectories and human ones were hardly revealed. Our results mark a strong milestone toward the full integration of robots in social environments.