Running to first.
 

In a coach pitch game Let's say a batter hits the ball and the pitcher fields it. The pitcher, instead of throwing to first or running to first base, he attempts to tag the runner. The runner stops running and takes a step backwards. The umpires then call the runner out (without being tagged). Reasoning is because the runner took 1 step back to avoid a tag. They said it is a Dixie youth rule that if the runner takes steps backward he is giving himself up therefore he is automatically out.

Was this the correct call?? Can it be found in the rule book or just a known rule?

Don't know LL rules, but it's likely a local rule, so you're on your own.

Don't know what the "Dixie Youth" rules are.

The typical core rules used by local leagues (ORB, NCAA, NFHS) don't have anything like that, but I believe it is used in fast pitch softball.

It's an organization akin to LL or Babe Ruth or Pony.

The rules are available on-line. The baseball rules are OBR based. I could NOT find a rule (in a one-minute search) that says that the BR is out for retreating -- so in the OP, the play should have been left alive. (I didn't check the softball rules.)

Of course, it's possible that the local league adopted some such rule.

It's a softball rule, but not a baseball rule.

Yes--softball rule.
In baseball he can back up but is out if he touches the plate.
You can find it in the MLBUM.