I post a lot about "game management techniques" and this guy sounds like he could use lots of help in that area.
Issuing warnings at the pre-game plate conference IS NOT an example of good game management! Trained umpires are specifically told NOT to use this approach.
When an umpire starts with warnings before the game has even started, it puts the coaches in a defensive mode and establishes an adversarial atmosphere. That approach will usually cause more problems than it will solve.
Any warnings should be issued during the game and should only be issued in response to a specific violation or sportsmanship issue. If an issue crops up during the game that requires a warning, deal with that specific issue on an individual basis.
Blanket warnings can also backfire and serve to undermine an umpire's credibility. Once the warning has been issued, if something related to the warning comes up during the game the umpire has painted himself into a corner. If he does not act with an immediate ejection, no matter how minor the infraction, he has shown that his words are hollow.
Conversely, if he follows up on his pre-game warnings, he is now put in a position of making an ejection for some borderline offense that could probably be handled in a much calmer, reasonable manner. Either way, the umpire's credibility is going to go down, not up.
Generally, an umpire has NO authority over anything that happens outside the fences, or involving anyone who is not a participant in the game. Fans are out of the umpire's jurisdiction. Fans can yell anything they want and an umpire should ignore it. The only instance where an umpire should act on behavior from outside the fences is if the actions become threatening toward a game participant or create a perceived safety issue.
In those cases, the umpire would be within his rights to temporarily suspend the game, putting both teams in their dugouts until the problem can be dealt with by tournament directors or school officials. But an umpire has NO authority to eject a fan or punish a team for actions by a fan.
(Some local leagues will give an umpire authority to deal with abusive fans, but that is a different issue than tournament or high school softball. Local leagues will usually own or lease the fields and thus have a different level of responsibility in managing the facility and game environment. Those "special powers" will be something specifically written into the league rules and are not powers granted to an umpire in tournament or school play.)
"Game management techniques" are not something the average fan, player or coach is usually familiar with. But when an umpire is lacking in this area, the results quickly become obvious. The game delays, arguments and ejections stemming from poor game management can turn a fun day at the ballpark into a painful experience for all involved.
An umpire needs to check his ego at the door before stepping onto the field, develop a thick skin to criticism, ignore comments from the stands, should not go looking for trouble and be able to handle tough situations without escalating them into an even bigger confrontation.