It is not technically an application for a stay, but it really is (for all practical purposes).
The United States this afternoon filed a motion with Judge Vinson of the Northern District of Florida asking him to clarify the practical impact of his holding. You can access the so-called Motion to Clarify here.
Here are the summarizing paragraphs, clipped from the first two pages of the DOJ's memorandum in support of the motion:
"The Court’s declaratory judgment potentially implicates hundreds of provisions of the Act and, if it were interpreted to apply to programs currently in effect, duties currently in force, taxes currently being collected, and tax credits that may be owed at this time or in the near future, would create substantial uncertainty. Because of the sweeping nature of the declaratory judgment, such an interpretation would pose a risk of substantial disruption and hardship for those who rely on the provisions that have already been implemented.
"Defendants will appeal both the Court’s judgment and the rulings that underlie it. This motion respectfully asks the Court to clarify the scope of its order, in particular that its declaratory judgment does not relieve the parties to this case of any obligations or deny them any rights under the Affordable Care Act while the judgment is the subject of appellate review, or, if the Court anticipated otherwise, to address specifically what the Court intends the parties’ obligations and rights to be under the judgment while appellate review is pending."
So the ball is now in Judge Vinson's court. He could effectively state that everything remains in place, and the ACA remains good law, pending appellate review. Or he could say that the United States is barred from implementing any aspect of the ACA, which would be the equivalent of an injunction. Or he could say something in between--that the United States is barred from implementing certain aspects of the ACA but not others (though it is unclear how such a middle ground could be squared with the substance of his opinion). And if he states that the United States is effectively enjoined from any sort of implementation, then the DOJ has clear grounds on which to seek a stay--or, more accurately, a clearer legal obligation that it can seek to have stayed.