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The contingency plans for the military are to scramble immediately and head out to protect the Ambassador and the others. Should they fail to get a response, they are to follow preestablished plans for rescue. Which means that a NO GO from the President was needed to stop the operation.

If the quick reaction force was already in country, then I would agree with you. If not, then it at least used to take a positive order from the Commander-in-Chief to cross the border unless that permission was in a standing order. I imagine the general had half his men coordinating the preparations for that operation and the other half combing through all relevant previous orders (and calling Sixth Fleet to see if the Navy had a loophole) to find something he could plausibly pretend counted as permission to cross that border in order to rescue American military and State Department personnel from armed foreign terrorists.