1. A gaze position estimation system, comprising: a camera that photographs a face of a user who is looking at a screen displayed on a display; at least one processor; and at least one memory device that stores a plurality of instructions, which when executed by the at least one processor, cause the at least one processor to: detect, from a photographed image of the camera, an eye area of the user and at least one of a face area of the user and a predetermined part area of the user, the predetermined part area being an area of a predetermined part of the user other than the user's eyes; obtain areal size information and position information of the eye area, and areal size information and position information of the at least one of the face area and the predetermined part area; and estimate a position in the screen that the user is gazing at, based on the areal size information and the position information, wherein the plurality of instructions further cause the at least one processor to obtain reference areal size information and reference position information of the eye area, based on areal size information and position information of the eye area when the user is looking at a reference position in the screen, and obtain reference areal size information and reference position information of the at least one of the face area and the predetermined part area, based on areal size information and position information of the at least one of the face area and the predetermined part area when the user is looking at the reference position in the screen, wherein the plurality of instructions cause the at least one processor to estimate the position in the screen that the user is gazing at, based on a first difference between the areal size information of the eye area and the reference areal size information of the eye area, a second difference between the position information of the eye area and the reference position information of the eye area, a third difference between the areal size information of the at least one of the face area and the predetermined part area and the reference areal size information of the at least one of the face area and the predetermined part area, and a fourth difference between the position information of the at least one of the face area and the predetermined part area and the reference position information of the at least one of the face area and the predetermined part area.