There’s often confusion between rebut and refute and both words may be misused when deny is most appropriate. It’s true that they all have to do with negation, but that’s where the similarity ends.
To deny a statement is simply that. If I say ‘all dogs are grey’, then if you deny or contradict that statement all you have to do is say: ‘No, they’re not.’
However, if you wanted to rebut my statement, you would have to provide a well thought out and reasoned argument that suggests that I might be wrong. This is used in debates, where speakers present evidence that supports a counter argument.
In order to refute my statement about dogs, you would have to prove it wrong, perhaps by showing me a dog of a different color.
Here are some quotations from publications around the web:
… Friday, Mr. Emanuel declined to speculate on whether the Trump administration would find another rationale to deny the city the grant — something that has never happened. Nor would he say if he thought that the administration … (www.nytimes.com)
… in which I addressed questions from Congressmen about my testimony before the global warming committee, I had already refuted in detail the points now belatedly raised again by the scientists who have written to Congress. (www.theguardian.com)
… opportunity to present a counterpoint to Mr. Wolff. And she has broadly discussed with friends the idea of a book to rebut “Fire and Fury.” … (www.nytimes.com)