It does not
As described under the Gaze universal monster ability:
Only looking directly at a creature with a gaze attack leaves an opponent vulnerable.
Not only that, but nearly all gaze attacks have a limited range (usually 30 ft), which must be met for the saving throw to be required. If your character is outside of that area, the gaze attack has no effect at all.
As for Scrying methods, those are treated as a separate sensory organ, and not exactly a replacement for your eyes. So, any effect that works through the eyes shouldn't work at all while using a scrying ability.
Scrying: A scrying spell creates an invisible magical sensor that sends you information. Unless noted otherwise, the sensor has the same powers of sensory acuity that you possess. This level of acuity includes any spells or effects that target you, but not spells or effects that emanate from you. The sensor, however, is treated as a separate, independent sensory organ of yours, and thus functions normally even if you have been blinded or deafened, or otherwise suffered sensory impairment.
However, reading the text of Fearsome Duplicate, I would say that your copy would get destroyed as soon as the creature looks at it if this gaze attack is one that does damage, such as the gaze from a Hound of Tindalos. But otherwise, a petrifying gaze, for example, shouldn't have much effect on it.
The duplicate immediately winks out of existence if it is hit by an attack or in the area of a damaging effect, or if it moves beyond the maximum range of the spell.