Not really. SCOTUS does not like new arguments made to them in an emergency hearing not brought below. It is beyond the pale. It simply is not done. But then there is a life hanging, twisting, slowly, slowly, in the wind, due to what I think might well be found to be legal malpractice. So it is one of those highly unusual situations. So there is hope. The last thing left in Pandora's box was hope. There is always hope.
Actually, the Supreme Court WILL entertain arguments that were not advanced below - as long as the actual causes of action that are buttressed by those arguments, are not new. So, if you brought a First Amendment cause of action, and supported it below with the argument that government unconstitutionally prohibited you from saying something, you could add, at the Supreme Court, an argument that government also FORCED you to say something - which is also a First Amendment violation. "Once a federal claim is properly presented, a party can make any argument in support of that claim; parties are not limited to the precise arguments they made below." Yee v. Escondido, 503 U.S. 519, 534 (1992); see also Dewey v. Des Moines, 173 U.S. 193(1899).
Problem with that principle in the Schindler case, is that the Supreme Court, I believe, is reviewing the lower court ruling (denial of prelim injunction) for abuse of discretion, so such a review might have to confine itself to the arguments that were presented to the court below - in order to judge whether its rejection of them constituted abuse of discretion. On the other hand, given the gravity of the life-and-death issue, the court might employ a more exacting analysis, asking if additional arguments for, say, a Due Process violation COULD have been propounded by the court itself but weren't (as a court might have been expected to do if the plaintiff is a non-lawyer representing himself).
I would never argue against Terri's express wishes to be denied nourishment, were that the case. But this whole thing stinks to the heavens. This is wrong. Horribly wrong. And I suspect that our nation will be irreparably harmed if this miscarriage is permitted to occur.