and Obama did not authorize the shoot. The Captain of the Bainbridge did, after noting that the hostage was in imminent danger. This has been part of the standard rules of engagement for my entire adult life. From the article you linked:
The pirates were pointing AK-47s at Captain Richard Phillips and he was in "imminent danger" of being killed when the commander of the nearby USS Bainbridge made the split-second decision to order his men to shoot, Vice Admiral Bill Gortney said.
I fail to see how a 'split second decision' in any way was made in the White House.