It's DM's call, but not by RAW.
Hypnotic gaze specifically says that you have to speak. Just like you can't just substitute telepathy for a verbal spell component, you can't substitute it here.
A DM might allow mental communication to substitute for spoken words, but the DM would also be justified in ruling that using Hypnotic Gaze constitutes an attack on the same grounds. I certainly wouldn't allow a Tempest Cleric, for example, to use Wrath of the Storm without breaking invisibility. I don't think that the language used in invisibility was intended to only apply to the Attack Action, but rather to the more general concept of an attack. In this case, I would rule that Hypnotic Gaze counts as an attack on the mind.