To those who say the SCOTUS won't take it, flat out, I say there are good arguments on both sides of that issue. The courts indeed get involved in separation of powers cases, but the threshold is not clear.
Purely hypothetical, but what if the Senate rules required unanimity for signing on to a treaty, in clear contravention of the Constitutional requirement for 2/3rds? That would effectively neutralize the Presidential power to make treaties. Similarly, in the case at hand. the President's power to approve Article III judges has been tipped.
Further, the Constitution provides for Senate oversight for Supreme Court nominees, and that the Congress can make laws prescribing the process for lower court judges (District and Circuit courts, in this case). Notice that the Constitution says the -mandatory- laws prescribing filling those lower offices may "... vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments." This clause intimates a limit on Senatorial power, and that for inferior Officers, the appointment power can be vested in the President alone, or in the courts -- but NOT interposing a higher barrier for appointment than the advise and consent of the Senate. It happens that the relevant statutes (for filling Disctict and Circuit Court vacancies) recite an advice and consent requirement, but the Constitution does not mandate this.
Clearly, if the Senate rules required unanimity in order to produce the advise and consent the Constitution requires for Supreme Court nominees, thereby permitting a single Senator to thwart the President in the exercise of his Constitutional power to fill a Supreme Court vacancy, we would have a Constitutional crisis. Would the SCOTUS take such a case? If yes, then we are talking about the threshold for taking a case. The Senate rules are now being used to effectively permit a minority of Senators to do precisely that, block Presidential appointments. The suit would be stronger if the President joined it, but I think the suit is ripe.
There is no doubt in my mind that the Democrats are perpetrating a Constitutional Crisis, and I sincerely hope they are called out on it.
As to whether or not the SCOTUS will take the case, I don't know. The point of this one-sided missive was merely to assert that the issue is legitimately arguable. It is not a cut and dried matter, and if the case is brought, the SCOTUS would be compelled to at least issue an opinion explaining why it is denying to become involved.