The award is just that. An award. He may choose not to accept it by leaving the field of play and being called out for abandonment... But to answer your actual question (may the batter continue batting?) - no... he may not.
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"Many baseball fans look upon an umpire as a sort of necessary evil to the luxury of baseball, like the odor that follows an automobile." - Hall of Fame Pitcher Christy Mathewson
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