This is one of the arguments made by Flynn in his opposition to the appeal (p. 8).
No rule or precedent authorizes a district judge to seek rehearing of
a mandamus order. A district court ordered to respond to a petition
for a writ of mandamus is not thereby endowed with the rights of a
party. The resulting mandamus from this Court to Judge Sullivan—just
like any appellate order to a district court—does not give the judge
standing to litigate issues “as a party, intervenor, or amicus.” Ligon
v. City of New York, 736 F.3d 166, 170 (2d Cir. 2013).
Judge Sullivan, the supposed umpire, does not make it to first base. He has no injury.
Of course, we'll have to see whether the courts agree.
This question (and answer by ohwilleke) raises the question of whether Sullivan can appeal. The Main Answer is, The Judge Can't Appeal; But Other People Can. Specifically,
The attorney appointed by the judge to present a position that the
Justice Department abandoned, might have standing to do so, but the
judge himself or herself, while listed as the Respondent in the case,
is only a nominal party and not a true real party in interest.
He notes that
A request for writ of mandamus ... is structured in the old fashioned
approach used in federal court as a lawsuit against a judge brought in
a court with supervising authority over the judge. But, in substance,
this is a legal fiction and formality used ... to provide review of
trial court decisions prior to the entry of a final decision on the
merits in a case ... Originally, a writ of mandamus really was a
lawsuit against a judge ... But, now that is just a formality and not
a real lawsuit in this context, and that has been the case in cases
involving writs of mandamus filed against judges for hundreds of
The fact that the judge's name appears in the ruling as a nominal party is the legal fiction here: the judge is not a party in this case.
Then, of course, to keep up with breaking news, as of today there is no longer a ruling for Sullivan to appeal: it was vacated from on high.