He used the right rule, but the wrong sentence from the rule. The rule is NFHS 4-1-4. The topic is "ball rotation." Here is the rule. I've bolded the parts that you are asking about.
Quote:
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The pitcher has a choice of balls at the start of each half-inning, unless both balls do not get put into play in the first half of the first inning. In that case, the pitcher in the bottom of the first inning must throw the unused ball. Thereafter, the ball in play should be returned to the 16-foot circle after every third out. The pitcher taking the field now has a ball to start the next half-inning. The pitcher may request the other ball before throwing a warm-up pitch by giving the ball in her possession to the plate umpire. The pitcher now has made a choice and must pitch that ball until such time as ti goes out of play or becomes blocked.
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Basically, once she starts her warm-up pitches, she may not request another ball "just because." The ball would need to be in unplayable condition (wet, damaged). Otherwise, she must continue to use it.