Sending additional forces into a foreign country always requires the president's approval. He was either absent, or refused "CBA".
Once the alarm is sent – in this case, from the consulate in Benghazi — dozens of HQs are notified and are in the planning loop in real time, including AFRICOM and EURCOM, both located in Germany. Without waiting for specific orders from Washington, they begin planning and executing rescue operations, including moving personnel, ships, and aircraft forward toward the location of the crisis. However, there is one thing they can’t do without explicit orders from the president: cross an international border on a hostile mission.
No administration wants to stumble into a war because a jet jockey in hot pursuit (or a mixed-up SEAL squad in a rubber boat) strays into hostile territory. Because of this, only the president can give the order for our military to cross a nation’s border without that nation’s permission. For the Osama bin Laden mission, President Obama granted CBA for our forces to enter Pakistani airspace.
On the other side of the CBA coin: in order to prevent a military rescue in Benghazi, all the POTUS has to do is not grant cross-border authority. If he does not, the entire rescue mission (already in progress) must stop in its tracks.
Ships can loiter on station, but airplanes fall out of the sky, so they must be redirected to an air base (Sigonella, in Sicily) to await the POTUS decision on granting CBA. If the decision to grant CBA never comes, the besieged diplomatic outpost in Benghazi can rely only on assets already “in country” in Libya — such as the Tripoli quick reaction force and the Predator drones. These assets can be put into action on the independent authority of the acting ambassador or CIA station chief in Tripoli. They are already “in country,” so CBA rules do not apply to them.
How might this process have played out in the White House?