Gaze aversion---the intentional redirection away from the face of an interlocutor---is an important nonverbal cue that serves a number of conversational functions, including signaling cognitive effort, regulating a conversation's intimacy level, and managing the conversational floor. In prior work, we developed a model of how gaze aversions are employed in conversation to perform these functions. In this paper, we extend the model to apply to conversational robots, enabling them to achieve some of these functions in conversations with people. We present a system that addresses the challenges of adapting human gaze aversion movements to a robot with very different affordances, such as a lack of articulated eyes. This system, implemented on the NAO platform, autonomously generates and combines three distinct types of robot head movements with different purposes: face-tracking movements to engage in mutual gaze, idle head motion to increase lifelikeness, and purposeful gaze aversions to achieve conversational functions. The results of a human-robot interaction study with 30 participants show that gaze aversions implemented with our approach are perceived as intentional, and robots can use gaze aversions to appear more thoughtful and effectively manage the conversational floor.