Babe Ruth game using OBR. R2 and 2 and 2 count on batter. Pitch is in the dirt and batter swings for strike 3 but R2 is thrown out stealing. Should the same batter start the next inning?

BR's time at bat eneded when he became a runner. The play is no different than"BR grounds to F6 who throws out R2 to end the inning."

The next batter leads off the next inning.

OK,
I can't imagine you don't already know the answer to this, but here you are.
Presuming less than two outs -
Follow the sequence of events:
Batter swings at a pitch for strike three - he is out;
Catcher throws baseball to base to put out base stealing runner;
Base stealing runner is out.

If these are outs number one and two, then the next batter in the batting order following the batter who struck out will appear at the plate.

If this is out number three, the batter who immediately follows the batter in the batting order who struck out will come to the plate in their half of the next inning.

The answer is the same, but I believe the poster's confusion probably lies in the runner being out number 3 (with BR making it to first).

The logic is the same --- if the runner out had been out number 2, would your batter still be batting? No - he'd be at 1st base. The batter's at bat was completed - he does not get another one.

OK
I get it.My apologies.
I was incorrect.

If retiring R2 is the 3rd out, then the inning is over and the batter who succeds the batter who struck out, will start at the plate next inning.

However ( and this is where I erred ), if retiring R2 is the 1st or 2nd out, and Batter/runner has advanced to 1st base, then B/R is safe and the next batter in the lineup comes to the plate.
Also, if retiring R2 is the 1st or 2nd out, and B/R gives up his right to 1st ( goes to the dugout, etc. ), then he is also out and the next batter in the line-up shall bat at the next time at bat for their team.

Got lost on that Pitch-in-the-dirt thing.
Egg on my face.
Oh well.

Here is the tricky part of a similar situation. Runner at second, 3-1 count on the batter, two outs. Pitch is a ball, runner is stealing third. Runner is thrown out. In this situation, the batter that was up walked, and the following batter leads off the next inning. That is why it is important to keep track of the count. Many coaches may send up the same batter.