It has been several months since I addressed this topic, but since I've got this draft sitting among my posts, I probably should just finish it and get it out there.
This part of the inquiry depends on the nature of the remedy sought in the Supreme Court. There's two possibilities regarding what Daniels was asking the Supreme Court to do when he brought suit: (1) to vacate the decision of the cabinet because Karen Hayes vote was invalid or (2) to ask for a judgment declaring whether Daniels can act as President. Because it's been a while since I saw that episode, I honestly do not remember what remedy Daniels was seeking. (Note: if anyone does, leave a comment and I'll change this). The bottom line is that option 1 is potentially a valid exercise of appellate jurisdiction; it asks a supposedly higher court to modify a decision of some other judicial or quasi-judicial body. Option 2, however, asks the court to do something other than an appeal or and something that is not auxiliary to an appeal. In other words, option 2 is an original proceeding, and it falls victim to the problems I've already addressed below. So, in order to keep this post going, let's assume that Daniels selected option 1.
Because we are in 24's version of the United States, we need to make a few more assumptions. Currently, in the real world, the appellate jurisdiction of the Supreme Court of the United States is limited by statute to the following cases: appeals as of right from a district court composed of three judges granting or denying an injunction, 28 U.S.C. § 1253 (2000), and discretionary appeals from final judgments of the federal courts of appeals, id. § 1256, and from the highest state court that a party could reach, id. § 1257. The cabinet is neither a district court of three judges, a court of appeals, or the highest court of a state, so, at least under current statutory authority, the Supreme Court would not have appellate jurisdiction over a decision of the cabinet regarding the President's suitability.
Statutes could be different in Jack Bauer's universe, however. Let's assume then, for the moment, that the 24 Congress enacted another statute granting the Supreme Court appellate jurisdiction over the cabinet when it decides whether a President is fit to discharge the office or not.
Given all these assumptions, it is still hard to determine with certainty exactly what is going on here. Is the cabinet like an administrative agency? If so, this would help. The courts of appeals exercise direct review over the decisions of some of the administrative agencies (i.e. without it being first handled by a district court). So, if the cabinet can be treated like an administrative agency (and I think it could), then the Supreme Court, like the courts of appeals, could directly review its decisions if statute so authorized.
Unfortunately, we are back to the original/appellate problem. What is the nature of the courts of appeals' review powers over administrative agencies? What if it is not appellate jurisdiction but instead original jurisdiction? If that were the case, then the Supreme Court would not be able to hear the case, as it would again fall victim to the problems enumerated below.
So, in the courts of appeals, are direct reviews of administrative agencies original or appellate matters? I have no idea, and I haven't been able to find any case dealing with this question (yet). I guess I'll have to stop here for now. Final answer: maybe?