The sentence is correct, but "avert" is not commonly used. Say "don't look away".

Using avert in this context is not natural. Avert = to turn away. It would be more natural to say "You're not looking at me." as a statement or "Why don't you want to make eye contact?" as a question inviting an explanation. Another phrase you could use is "Don't take your eyes off me!" as a strong request or demand that you remain in the eyesight of the listener.

The correct phrase is " To avert one's gaze " ,,, we usually only "avert" a "gaze".