Not exactly. True... the 12 mile limit extends into any navigable water. Authority to board and inspect insde that line is absolute. That's just the way it is. The CG also can board any U.S. flagged vessel anywhere in the world, in international waters. Or... can board any vessel flagged in a country for whom we have a treaty granting authority to the USCG to conduct boardings. Treaties to this effect are in place with regard to most countries of So. America.
Any vessel in the Pacific or Atlantic between Cape Horn and Canada can be searched, regardless of flag. Period. It is done. And I'm not sorry about it: we're not patsies.
One small correction, the USCG can board any vessel in international waters no matter the nationality to verify paperwork and to ensure the vessel is not stateless. However the boarding stops when this has been completed and there is not any searching of the vessel allowed. However this means that if you run and ignore orders to stop you can be stopped by force. Now mind you this is not done very often with foreign flagged vessels but the right to do so is there.