An electronic device may have a display and a gaze tracking system. Control circuitry in the electronic device can produce a saliency map in which items of visual interest are identified among content that has been displayed on a display in the electronic device. The saliency map may identify items such as selectable buttons, text, and other items of visual interest. User input such as mouse clicks, voice commands, and other commands may be used by the control circuitry in identifying when a user is gazing on particular items within the displayed content. Information on a user's actual on-screen point of gaze that is inferred using the saliency map information and user input can be compared to measured eye position information from the gaze tracking system to calibrate the gaze tracking system during normal operation of the electronic device.