This happens at my house. The dog won't even look at the cat.
Ours too. If he looks at the cat, it's only out of the corner of his eye and he sits very still, like, "Oh no! I hope she didn't see me looking at her!"
Our cat is a bit of an instigator though and will chase the dog into the living room to get him in trouble since she knows he's not allowed in the living room. (He's a German Shepherd and is outside most of the day and he stays in the mud room when he's inside.)