Is a Re-Touch Required?

[cshs81]

R2 and R3.

Groundball. R3 heads for home but stops when the catcher catches the ball.

R2 has touched third base and rounded it when he sees that R3 is heading back towards 3rd base.

R2 does not re-touch 3rd base as he starts back towards 2nd(note: he's now between 2nd and 3rd a few steps off of 3rd base).

However, F2 overthrows 3rd base in an attempt to throw out F3. F3 scores.

F2, seeing the ball in the outfield and that R3 has scored, turns and heads towards home without touching 3rd base.

Is re-touching required since he had already touched third AND since he did not re-touch it on his way back?