Pixar.wikia:
He tries to hold her hand as he had seen in the film, which he determines to be an act of sharing love. Unfortunately for him, EVE does not understand what he is trying to do and thus does not return his love, especially as she is distracted by the mesmerizing flame of a match lighter and does not fully appreciate the film as he does. WALL•E then decides to show her the plant he found. Suddenly, EVE scans the plant, and her system automatically initiates a shutdown mode, which takes the plant and stores it inside her torso, sealing up into her pod form and shutting down. WALL•E does not understand why she will not unlock or speak to him, but nonetheless does everything in his power to keep her motionless body safe from the elemental hazards of the desolate Earth in the hopes she will wake up, taking her out on a series of "dates" in the process. Eventually WALL•E gives up hope that she will ever awaken and leaves to resume his work. Soon the ship EVE arrived in returns and takes her away. While it is clear that EVE had sent a homing beacon to her ship to collect her, WALL•E is unwilling to leave her and climbs onto the side of the ship, telling Hal to stay where he is, clinging on as