From Wikipedia: Air quotes: Air quotes, also called "finger quotes" or "ersatz quotes" are virtual quotation marks formed in the air with one's fingers when speaking. This is typically done with hands held shoulder-width apart and at the eye level of the speaker, with the index and middle fingers on each hand flexing at the beginning and end of the phrase being quoted. The air-quoted phrase is—in the most common usage—very short, at most a few words.
Depending on the context, I might say he "uses air quotes," although I wouldn't be too surprised to hear "the man air quotes" (although I've never heard it used as a verb). I would not use "gestures/mimics."