The umpire did exactly what he is supposed to do. He never made a call so he didn't overturn. I am assuming that he did not signal "safe" with his hands - as that would not be normal unless there is a close play.
When a player misses a bag and is either in the dugout or on another bag there is no ability for the defensive team to tag him out. These plays must go through an appeal process much like a defensive player asking for the home plate umpire to appeal a check swing to a field umpire.
You understand if the defensive player touches the plate, it means nothing right? The player could run towards the dugout to tag the player (but doesn't have to). At that time after I saw the player move to where ever the player was, if that player did not take a direct route to home plate to retag it, you would call them for running out of baseline. Really the only way they could be safe is if the defensive player dropped the ball during tag.
The tone of your question displays an annoyance that this happened. This is one of the few areas of baseball where an on the field event can be directly affected by coaching. A good coach would have seen his player miss home plate - one of the coaches should be looking at this for 9-year olds (and they should know better at that age). That player could retreat at any time back to retag home until the umpire calls time or they actually step into the dugout (this is a bit controversial). At the same time the defensive manager/coach should have told his player to go tag yours. Either way your team was outcoached. It is a runner error if he misses and tries to go back, coach's error to not pay attention.
However in this process for an appeal once the offensive player goes to dugout - is for the pitcher to step off the mound and throw to the bag(s) in question after ball is dead. If done correctly the pitcher should have called time, walked to mound, stepped off of it, and then threw to the catcher. The umpire at this age should guide the player to do a correct appeal - really you have unlimited time to appeal as long as you don't throw another pitch, so there is no getting around making the call. At the same time as a coach for 9-year-olds you probably have a junior umpire - coaches should be helping them explaining what they are appealing.
The fact is this is a good question to have at the moment. It is troublesome that you had to go to this site to get an answer. Your coach should have explained to the parents that the other coach appealed the runner scoring and why the runner was out. Not "whoa is me, how can the umpire do that?!" This bit of ineptitude in coaches, especially at a youth level not only hurts the team but creates negativity in player's attitudes towards umpires/officials.