Yes! Really! The word has legitimate meaning in principle. It would mean to simultaneously refute and repudiate something, such as a political position. But here's the problem: Palin, who coined it, misused it! The correct word was repudiate! Perhaps she realized that and, doubling down on her mistakes [as she always does] she used the word "refute" in its place in a later tweet! WHAT THE FOX!
So let's boil it down! here are the relevant definitions from Merriam-Webster:
refute, transitive verb, 1. to prove wrong by argument or evidence : show to be false or erroneous. 2. to deny the truth or accuracy of
repudiate, transitive verb, 1. to refuse to have anything to do with : disown. 2. a : to refuse to accept; especially : to reject as unauthorized or as having no binding force. b : to reject as untrue or unjust
So, they are similar and related, and repudiate can often take the place of refute, but there is an important distinction. One is about logic and fact whereas the other is about position and opinion. So this confusion is more than a misuse of English or the coining of a new useful word. In the case of Palin, it was a Freudian slip. Like many people poorly educated in epistemology, she has a fundamental inability to separate fact and logic on one side from opinion, belief, and feelings on the other. I'm not going to claim that I'm perfect. People naturally have their judgment clouded by opinions and beliefs, but the ability to distinguish between them in principle is vital for handling complex and controversial issues. What we need in a leader is the ability to cut through the noise and strong feelings that dominate such issues to find the core, and then calmly and rationally explain it to both sides. Palin, more or less in every single case, just joins in with the rabble.
This "folksiness" is precisely why the "liberal elite" dislikes Palin so much, but also why her fans love her so much. One can refute her positions all day long, but her fans she will never repudiate her -- refuting Palin only makes them love her more, precisely because they don't know the difference between refute and repudiate!