To gawk at someone means to stare at them too much, to the extent that it is inappropriate. There is a hint that a person who gawks looks a bit stupid while they are doing it as well. It is a negative thing. Gawking at one's self would be staring at yourself in the mirror so much that other people who see you wonder what is wrong with you, that sort of thing.

The phrase "Stop gawking at yourself" means stop staring /Looking at yourself