Can checked swing be a 'fourth out' appeal?

[Chukinrox]

This could make a huge difference in the game. If the batter DID indeed swing at the pitch, that would be the third out and there would be no chance at a play at second. Either way the inning ends. No difference? Not exactly. Say the batter is a great hitter. If he was struck out, then the runner on second would be stranded and the next batter would lead off next inning. If you give the defense the out at second and do not allow them to appeal the check swing, then the same great hitter would lead off next inning. Now the only concern I have is "does an appeal of a check swing have to occur before the defense makes a play at second?" Something worth pondering. I don't think it is fair to not allow the defense to appeal...