From what I understand, typically, in situations like that, the team informs the umpire of a potential injury first, and then removes the pitcher (usually after talking to the pitcher). In this case, the Red Sox signaled for a pitching change (as if it was just a normal pitching change), and then went to the umpire to inform him of the injury.
Because of the order it was done in, Girardi didn't believe that Delcarmen should have been allowed more than the normal eight warm up pitches. I don't think the Red Sox did anything wrong, but it's a reasonable protest in my opinion, because it's not the way things are typically done.