I would argue that the author did not cause the PR disaster, Google did. They author submitted the memo directly to the diversity team months ago and they ignored it. It was posted and updated on Google provided employment forums for months and Google ignored it.
They are now publicly vilifying him for expressing opinions that are now being publicly supported by scientists. They have essentially publicly attempted to silence him because they don't like his opinion, validating his initial complaints.
I don't know if Pichai should be fired but a lot of people dropped the ball on this and they have escalated a story that could have been quietly handled in house via a few conversations months ago.
I have no idea if Damore's arguments for biological differences are valid or not, some scientists have stated they are (mileage may vary) but I don't feel that anything he said was stated with malice of with the intent to denigrate anyone. He may not have had the best communication skills but he was trying to start a conversation not a war. Google for some reason responded to his inquiry with the equivalent of scorched earth and are now realizing that perhaps they overreacted.