Actually, ROE is determined at very high levels, and has NCA approval before it can be implemented in the field. Like everything else, the ROE is stated very broadly at the NCA level, and then defined more specifically at each lower level to conform to the intent of the CIC's approved ROE, while providing clear guidance to the warfighter. So in a broad sense, the President did approve the ROE, and his staff put it together.
That said, once the ROE is stated, field commanders should not need to ask higher ups for further approval to execute, unless such is required by the ROE.
As you state later, the ROE approved at that level is extremely broad to the point of being more an expression of the CIC's intent than rules on the execution of that intent. The NCA does not dictate tactical level rules of engagement. They don't have the visiblity required to micro-manage at that level in all the theaters we currently operate in.