Since NoiseGator very simply takes an audio input and applies a software gate to it, there's no way to tell Skype or other software to use the newly filtered audio. To help us accomplish that, we need to use either VBAudio for PC or SoundFlower for Mac.
These are just virtual drivers. You do not need to use or configure the software once they are installed.
With NoiseGator open, choose the microphone you are using for input, then choose your "virtual" audio as your output. On PC, it will be something like "CABLE Input (VB-Audio Virtual Cable". On Mac, it will be "Soundflower (2ch)".
You should also experiment with the settings, especially the threshold since that's the main point of the noise gate. Find a setting that blocks out background noise but keeps your voice very clear.
Choose your virtual audio output as the input for your software. Now your software thinks this interface is a direct audio input but its actually routing through NoiseGator first!