whale eye
n.— «When a dog turns his head away from you but his eyes are turned as far to the side as possible in your direction, the whites of his eyes take on that crescent shape. When a dog’s eyes take on this appearance, professionals in the field refer to it as “whale eye.” This visual sign is thought to occur because the dog is so afraid that he is not looking at you, but he’s too afraid to take his gaze away from you entirely. » —“The Eyes Have It —what Can Be Seen In A Dog’s Eyes” by Karen London Dog Behavior Blog May 12, 2008. (source: Double-Tongued Dictionary)
Further reading
In 1968, students at Cheyenne High School in Cheyenne, Wyoming, compiled a collection of their own slang, including the word Clyde, used to refer to one’s head, as in Use your Clyde! This is part of a complete episode.
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The “whale-eye” you mention is really called the “baleful” look. It doesn’t indicate fear, at all. In fact it is a really coy way for a dog to tell you that you have disappointed him, missed something, or generally have failed to be “the kind of person your dog thinks you are.” We call it the “white-wall tire” look when our Black Labrador retriever discovers something amiss in his routine daily expectations!