This study investigated the modulation of center of pressure (CoP) displacements of young adults as they performed predictable and unpredictable saccadic eye movements in stationary and moving visual scenarios. We also examined whether the relationship between CoP displacements and visual stimulus, provided by the moving scenario, and gaze control itself, are affected by the demands of the saccadic tasks. Fifteen young adults (20.53 ± 2.1 years old) stood upright on a force plate, inside a moving room, wearing an eye tracker while performing three tasks: gaze fixation (fixating on a target in the center of the screen), predictable task (saccades while following a target which continuously appeared and disappeared on the right and left sides), and unpredictable task (similar to the previous task, but the participants did not know which side the target would appear on). For saccadic tasks, the target appeared at a frequency of 1.1 Hz and with eccentricity of 11.5 degrees of visual angle. Two blocks of six trials were performed: in the first block, the room remained stationary and in the second, it oscillated (0.6 cm amplitude; 0.2 Hz frequency). Mean amplitude of CoP displacements was lower in the saccadic tasks compared to the gaze fixation, in both conditions; and higher in the moving scenario than in the stationary condition. Variability of CoP displacements was lower in the unpredictable saccades than gaze fixation task. Saccade reaction time was longer in the unpredictable than predictable task. We conclude that CoP displacements are reduced to facilitate performance of the saccadic tasks regardless of conditions and task complexity. Furthermore, lower variability suggests modulation of CoP displacements to deal with the increased attentional demands associated with the performance of the unpredictable saccades, indicating the important role of visual task constraints in postural control.