You can sue the government for the actions of merchant vessels that it owns or operates under the Suits in Admiralty statute of 1920. That statute was enlarged to allow you to sue against the actions of naval vessels of the government in the Public Vessels Act of 1925. Such suits are brought in Federal court only under Admiralty. There is no in rem principle involved with public owned vessels.
On the high seas you have a right to free passage and you may even have this right within the territorial waters of a nation if those waters involve a sea lane normally used for international passage-most notable in the waters of archipelagos.
You are not required to respond to the commands of a foreign naval vessel nor be boarded by the members of that vessel's complement when within international waters. There is a caveat to that law though. If the nations of the warship and your vessel's flag have an agreement allowing such actions your government has effectively waived such immunity. These agreements are more common than might be supposed.
There are any number of reasons to be diverted from a naval exercise area under various laws. The most common that is run into on the high seas is that you must give way due to the naval vessels being restricted in their ability to manoeuver under the Colregs.
If you are boarded or shepherded out of a naval exercise area against your will your options are limited to those against the flag nation of the warship and within your own country's courts/government. Assuming that there is no collateral agreement between the two flag nations you may be able to proceed at Admiralty but you are going to have to show a real loss, not just a potential loss. As to your being boarded by a foreign nation's armed forces you will have to rely on your own government to pursue such a case assuming you sustained no documented losses. Generally speaking, these types of situations occur when Greenpeace or some other group intentionally tries to interfere with a nuclear test or something and the country doing the testing removes them from the designated area. In the cases where the Greenpeace vessel was US flagged they've not had much success with the US pursuing their grievances against the boarding nation. There is a fairly strong presumption that such boardings or shepherding of vessels out of areas of naval operations is for the vessel's own safety.
The short answer to Cam's specific question is that there is no law, there may be a treaty, that allows the warship to so direct. To the best of my knowledge there is no French requirement, as there would be no US requirement, to obey the orders of the naval vessel. Of course, as a matter of practical consequences you are assuming all risk for your actions and it would be highly unlikely that your government would offer much in the way of support.
Outside of territorial waters you are free to ignore the entreaties of any naval vessel not of your flag. You're also free to bear the consequences and perhaps the unwillingness of your own government to make it into an international incident.