In the absence of a living will or an advance directive from the President, spouses are usually the ones who get to make medical treatment decisions. If a spouse &/or children don't exist or aren't available, then - depending on the jurisdiction - a "close friend" familiar with the person's values (i.e. possibly the Vice President/Acting President in this scenario) can make the decision, & some jurisdictions have developed mechanisms for "unbefriended" patients, usually involving choices by designated physicians (e.g., the President's doctor, the official physician to the President, etc.) often in conjunction with other physicians (presumably a medical team) &/or ethics committees.
The question though is whether there isn't an advanced national interest at stake, and the First Lady or First Gentleman has no jurisdiction to make such decisions. I guess it could be tested before court if a disagreement were ever to occur.