Kirk isn't going anywhere for at least 5 more seasons.
Like others have stated, if fans really want to push the issue, the only effective way is to spend those dollars elsewhere. Don't go to the games and don't purchase the merchandise, and the deep-pocketed boosters have to withhold donations. I'm not necessarily advocating for this, but, to do it quickly, there would have to be some type of concerted social media organized boycott.
As long as there is money and fan support, Barta will sit on his hands. That's the bottom line for pretty much every major program in the country.