Ya the guy had the authority. He has still tarnished himself I suspect - forever.
Well, not forever.
His authority, notwithstanding, it was his OBLIGATION under Canon Law to do what he did. Failing to do what one is obliged to do because of human respect is, plainly put, sinful. I would think he wasn't sitting at his desk deciding this action based upon how he would fare in a popularity contest.
It has been said that one of the greatest curses today is human respect – the vice that makes one compromise to gain other people’s approval. One's opinion about whether Archbishop Burke "tarnished" himself in the eyes of others by performing his duty as the lawful shepherd of the Church, in the end when he is judged, may be a good thing - particularly if one recalls the 8th Beautitude.