Yes, it has been done on occasion, Justice Brown most recently, but I am not aware of any federal statutory requirement covering the SCOTUS which makes its own rules. As you are aware it is a co-equal branch of the government so if it had to answer to federal statutes, it would not be co-equal.
Second, he did not take part in a deliberation on the case, just as your post states, he is the judge who oversees the Eight Circuit.
From your original post:
"On Saturday, Justice Thomas, who oversees the appeals court whose ruling is at issue, ordered prosecutors to respond to the application by Thursday. Such a request for a response is almost always a sign that the full court will weigh in on the matter."
So that looks to me as if his ruling on Sen. Graham's emergency application is that the prosecutors can and should respond and likely the entire Court will take up the matter.
Oh, Halleliua, Halleliua
Throw a nickel on the grass–Save a fighter pilot’s ass.
Oh, Halleliua, Oh, Halleliua
Throw a nickel on the grass and you’ll be saved.