Warning or ejection?

Why would you warn or eject him? His behavior wasn't directed at you or another player. He's unhappy with his own performance. Let the coach take care of it. There's no need for your involvement at all.

Your
job as an official is to control the game. Controlling the game means
controlling the participants: players and coaches.

It’s
impossible to answer your question directly. You were there and I wasn’t.
That’s not meant to avoid your question. It means that only you can interpret
what happened; the situation, the heat of the moment, the attitude of the
coaching staff. With experience this gets easier so don't give up. I
appreciate that you wanted to officiate in this situation. It’s never good to
bury your head in the sand and take no action at all.

You
might find that coaches will take care of that situation without your involvement.
Good coaches are good teachers and they won’t let that behavior continue. But
sometimes good  coaches don’t see the bad behavior and don’t know a
correction is needed. In that case, if you know that you have a teaching coach
(e.g. one that doesn’t put up with that behavior), it might be best to just to
talk to the coach between innings.

In
high school that action would earn, at minimum, a warning. So why would a good
official let this happen at a younger aged contest? You’d be setting the
expectation for the player that this is acceptable behavior only to have them
learn the hard way down the line that it’s unacceptable. And if your advancing
as an official, the problem you create today might be YOUR problem
tomorrow! From my experience, someone that wants to ignore that that
behavior is going on may be letting a lot of other unacceptable things happen
because they don’t want to be confrontational. The only time I’ve had problems
on the field/court was when I was lazy or too lax. You’re absolutely not, and
should never try to be, the star of the game. But if you step on the
field every time with a consistently respectful and authoritative demeanor
you’ll garner respect that will precede you. 

Finally,
to give you my best answer to your question, any action is better than no
action.

-Troy

p.s.
What would you expect Coach John Wooden to do in that situation?

You must take in consideration, these youngsters are 11-12 yrs. old, parents and friends watching, and they do behave wrongly, but, as Mr. Ballard put it, our job is to officiate, using all means to control the game. We have been taught to use "Preventing Umpiring", to avoid any problem that could ensue later in the game. I, for one, will talk to the coach, (not the player), and would ask him to take care of this situation, not as a warning, but, to convey to the player that his behavior is not acceptable. Remember, my fellow officials, we can teach these players at this age, to show respect for the game, and any action taken, is better than no action. 
Manuel Provedor