Can checked swing be a 'fourth out' appeal?

[Snorg]

Long-time reader, first-time poster (a lurker, I suppose). I thoroughly enjoy the forum and the banter, and have learned quite a bit (bad knees have put the kibosh on my umpiring for the last few years, so I may be a little rusty, but I still have the passion). Anyway, here's my sitch.....

Runner on 1st, two outs, two strikes on the batter. On the next pitch, runner breaks for 2nd, batter check swings, and catcher's throw retires runner at second for the third out. The question is, can the defense now appeal the checked swing and (assuming the BU rings him up) take the 'fourth out', retiring the batter so he doesn't lead off the next inning?

I didn't see it, but my brother called me about a similar play that happened last night in the CLE-BOS game...except the runner was safe at 2nd. The checked swing was appealed for the 3rd out of the inning, so no stolen base was awarded in the box score. That's what got me thinking......

Thanks.