Safe/not safe at home/no call made

[Coach B]

A runner seemingly scores at home with no throw to the plate, but does not actually touch home.  The defense assumes he scores, and gets the ball to the pitcher and prepares for the next batter.  The runner (who "scored") simply walks to his dugout, and the next batter gets ready to hit.  The plate umpire says nothing and never signaled a safe call, and never signaled out as the runner walks off.  Everybody in the park thinks he scored.  What's the call?  

[Velho]

17 minutes ago, Coach B said:

What's the call?

There is no call unless and until the defense appeals that the runner missed home plate. That appeal has to happen before the next pitch (or play depending on the ruleset).

[Coach Carl]

A runner is assumed to have acheived a base when he passes it, even if he does not tag it.  The defense must appeal the miss in order to get a ruling from the umpire (if missed, the umpire calls out, if tagged, he calls safe).

An umpire should be careful not to tip off either team of the miss, if he sees it.  It is the defense's responsibility to see it and appeal.  It is the runner's/offense's responsibility to notice and try to correct the error before the appeal is made.

I'll let my umpire friends correct me if I am wrong, but with no play at the plate you shouldn't get a safe signal.  Since the defense did not try to tag the runner, there is no call to make.