Since dogs don't use words to express their feelings, reading their signals isn't always a clear path to enlightenment. Because of that, understanding the nuances of their expressions and gestures is imperative. When a dog refuses to make eye contact with another, he's probably feeling uneasy about the situation at hand.
If a dog won't look at another dog, then you can make a safe bet that he's feeling awkward and concerned about something. He is nervous about the other pooch's presence, and doesn't feel at ease with the idea of dealing with him directly. By abstaining from eye contact, the dog doesn't have to go forward with any type of communication -- a safe haven of sorts. This dog is definitely not feeling too self-assured at the moment.
Avoidance of eye contact can indeed signify unease, but can also signify full-blown terror, as well. If a dog is terrified by what another dog might do, he may not only keep his gaze away, he may also display other clear indications of apprehension -- think a cowering body position, squinting eyes, shivering, tail hidden between the hind legs, tightness of the mouth area, whimpering, exposed teeth, stiff muscles and perhaps even involuntary urination.
If a dog keeps his glance averted away from another canine even before the atmosphere gets tense, then he may just be sending a message of submissiveness. He wants to maintain the tranquil environment, and is avoiding looking at the other dog as a way of expressing humility. If a dog feels subordinate to a dominant dog, keeping his eyes away can be an effective way of communicating that point -- and staying away from any potential fights. "Don't look at me for a fight -- I'm lower than you are."
If, on the other hand, a dog looks straight into another canine's eyes, he means business -- uh oh. He feels sure of himself and is being extremely forward in his interaction. By gazing straight into another dog's eyes, the dog is essentially sending out an invitation to go to battle. He's provoking the other party by "saying" something along the lines of, "Oh, so you think you are tougher than me? Not if I have anything to do with it."