In recent years, the number of bedridden patients, including amyotrophic lateral sclerosis (ALS) patients, has been increasing with the aging of the population, owing to advances in medical and long-term care technology. Eye movements are physical functions that are relatively difficult to be affected, even if the symptoms of ALS progress. Focusing on this point, in this paper, in order to improve the quality of life (QOL) of bedridden patients, including ALS patients, we propose a drone system connected to the Internet that can be remotely controlled using only their eyes. In order to control the drone by using only their eyes, a control screen and an eye-tracking device were used in this system. By using this system, for example, the patients in New York can operate the drone in Kyoto using only their eyes, enjoy the scenery, and talk with people in Kyoto. In this drone system, since a time delay could occur depending on the Internet usage environment, agile operation is required for remotely controlling the drone. Therefore, we introduce the design of the control screen focused on remote control operability and human eye movements (microsaccades). Furthermore, considering the widespread future use of this system, it is desirable to use a commercial drone. Accordingly, we describe the design of a joystick control device to physically control the joysticks of various drone controllers. Finally, we present experimental results to verify the effectiveness of this system, including the control screen and the joystick control device.