R1 on first, no outs. Long, caught fly ball to F8. R1 reaches second base, goes about 1 foot past second base, but does not touch it, he then retreats back to first base without touching second.
Must he touch second on his way back to first, even though he never touched it around the first time.
Would he be subject to a legal appeal at second?

Yes. He must touch the base the "last time by." Had he only touched it on the way back, he would not be out on appeal.