Ah, what an exquisite pseudo-legal question.
First, I have to second @zxq9 's objection; if he's totally impervious to any conventional weaponry, then US military functionally are children throwing rocks at him. But lets say that it's the intent that counts, or that maybe he's not totally impervious and could come to harm if he just let them attack without defending himself at all. Then there is a couple of salient points.
Surrender and Knightly Code
To start with, I find it rather uncharacteristic of a knightly code to contain a stipulation against surrendering. This would be the case for codes of honor among the honor-before-reason Proud Warrior Race guys (think Spartans), where their conviction of their own absolute martial superiority would be central to their self-image, so surrendering would be unthinkable.
Knightly codes (such as they were and what there was of them - keep in mind that what we have today are heavily romanticised and idealised versions that almost nobody actually followed) were codes of ethics as well as codes of honor. Knights would usually be expected to fight other knights and it was well understood that sometimes God, or fortune la guerre might favour the enemy. In such cases, while outright cowardice was absolutely condemned, simply acknowledging defeat by surrendering would be perfectly acceptable.
In other times, overriding ethical concerns would force the knights to surrender without even trying to fight; example of this is the famous case of Johannites (aka Maltezian Order) surrendering Malta to Napoleon without firing a shot because they considered it unacceptable to fight fellow Christians.
But let's accept that for the sake of the argument and see what we can do about that.
The Knight's Perspective
Assuming that the Knight's Code was designed with realistic warfighting in mind, even if it precludes unconditional surrender, it is sure to permit other means of ending the conflict.
Fortunately, all the various Laws and Customs of War have facilitating the restoration of peace as one of their primary purposes. The knight should thus be permitted to use some means of negotiating the end of the conflict; in history these would be variously called offering terms (which has a wide variety of meanings, and could include anything from total surrender of one side to some sort of compromise), suing for peace (which is a negotiation to end a conflict before its definitive military resolution either way), or offering white peace/status quo ante bellum (essentially let's stop shooting and pretend this never happened, War of 1812 and the Indian part of the War for Austrian Succession were concluded this way).
If he is familiar with some of the basic customs, he could try brandishing a Flag of Truce, riding up to someone responsible and offering terms under which hostilities would end. Depending on how he's feeling, and what his Code requires of him, this could range from simple cessation of hostilities (the white peace option), or he could demand some sort of satisfaction as war reparations. If the other side accepts, he has concluded the "war" honourably without surrendering.
In a situation where most of the might of the US military has already been tried and found impotent, there will immediately be powerful political desire to end this quickly, preventing further casualties and embarrassment.
Can the Knight present himself as a representative of a foreign sovereign power? If so, then this might be the best solution for the conundrum.
If he deigns to negotiate for terms and presents himself in such a way, then US is in fact in breach of international law, because it has engaged in an act of war without a formal declaration thereof. Furthermore, this is probably also a violation of the US Constitution - I am not an American, but I seem to recall that only a vote of Congress can authorize a declaration of war or other military action against a foreign sovereign power.
This is sufficient justification for the US to acknowledge their mistake, agree to cessation of hostilities and offer whatever reparations might fit the satisfaction requirement. And to do so quickly, before this becomes an even bigger political disaster.
The political fallout is going to be tremendous; heads are gonna roll, metaphorically speaking. Expect whatever military commanders authorized the initial attacks to be court martialled and called to task for the loss of life. The cabinet might catch some heavy flak for allowing the whole thing to happen in the first place.
And if the Knight doesn't call himself a representative of a foreign sovereign power? Well, the US might just unilaterally declare him that - as a legal fiction - in order to save face.