???
I don’t see any reason why not.
It would be a good way to take himself out of office because after he failed confirmation by the Senate the VP and the Cabinet would invoke the 25th Amendment and declare him incompetent for the presidency. 
Why do you assume he wouldn’t be confirmed?
I can see a lame-duck President in his last term successfully appointing himself, if an opening came up and he was generally perceived as qualified. Clinton probably could have gotten away with it.
I don’t know why he’d want to, though. I think a Carter-esque ex-President could do much more good in the world than any one Supreme Court justice. Only one former President has served on the Supreme Court.
If the president was confirmed he would have to resign in order to accept the new appointment as the constitution has that pesky bit about separation of powers.
I do believe that by the time Clinton’s second term was ending, his eligibility to even argue a case before the Supreme Court (an honor more than anything) had been rescinded, so I somehow think being APPOINTED to the Court would have been trouble, aside from the fact nominees need to be approved by Congress.
I don’t know what you mean about “eligibility,” since the Constitution lists no official qualifications for serving on the Court. The President can pick anyone he wants.
He needn’t be a member of the Court’s special bar.
He needn’t be a member of ANY bar.
He needn’t even have a law degree.
He needn’t even be an American citizen, presumably.
Legally, there is no reason why he could not nominate himself – although the timing could get a bit tricky.
In terms of precedent, one would have to go with Taft, the only ex-President Justice (and he was Chief Justice). Although it was an open secret that he wanted to be on the Court more than being President, he declined to nominate himself, feeling it inappropriate to do so, and had to wait until Harding nominated him in 1921, nine years after leaving office.
If it were, say, five days before the end of the term, and the President nominated himself, knowing that the confirmation hearings wouldn’t even get started until after he left office, would the incoming President have the right to rescind the nomination and make his own nomination in its place?
Or even more loony, if the new President didn’t like the old President, would the new guy be able to nominate his own candidate, and require Congress to choose between the two nominees?
Yes. The president can withdraw a name from consideration for an appointment; that’s a power inherent in the office, so whoever holds the office has the power.
As to your second question, I doubt it. There’s either a nominee or there isn’t.
As Fiver notes, there are no eligibility requirements to serve on any federal court except appointment by the president and confirmation by the Senate. The only officer in the government who is required to be “learned in the law” is the Solicitor General.
–Cliffy
Cite?
I’m really not being facetious. The constitutional separation of powers is an implicit doctrine, which is abundantly evident throughout the Constitution and statutes, but it is not spelled out anywhere in the form that an outgoing president “would have to resign in order to accept the new appointment.” (Several early state constitutions, particularly the Massachusetts constitution that John Adams drafted, did contain an explicit provision to the effect that no officer serving in one branch could simultaneously exercise an office in a different branch. But the U.S. Constitution of 1787 does not.) The Constitution itself contains counterexamples: The Vice President is both the Chief Executive’s successor, and the presiding officer of the upper legislative chamber (and while the same individual would not play those two roles simultaneously, most modern Presidents have treated their Vice President as at least a de facto member of the administration). The Chief Justice presides over the Senate when it tries a Presidential impeachment.
There are a few particular constitutional restrictions on certain offices being incompatible – for example, “no person holding any office under the United States, shall be a member of either House during his continuance in office” (art. I, § 6), and “no Senator or Representative, or person holding an office of trust or profit under the United States, shall be appointed an elector” (art. II, § 1) – but there is no general constitutional prohibition on simultaneously holding more than one federal office or holding offices in more than one branch. For example, the Vice President could also head a Cabinet department; when the Republican party nominated Ronald Reagan in 1980, some news outlets reported a tentative deal with former President Ford in which Reagan would name Ford both as his running mate and, if elected, as Secretary of Defense. And federal judges – including sitting Supreme Court justices – have accepted temporary commissions from the President for diplomatic and other missions.
The most significant historical example of an executive officer (admittedly not a president) serving simultaneously in a judicial office is John Marshall, who served as both Secretary of State and as Chief Justice for several weeks:
From Biographical Directory of the United States Congress.
From Legal Information Institute, Biographies of the Justices.
So, does this mean that John Kerry could theoretically retain his position in the Senate if he wins the Presidency?
No. The Presidency is an “office under the United States,” and falls under the explicit textual prohibition that “no person holding any office under the United States, shall be a member of either House during his continuance in office.”
Could the Vice President while explicitly acting as President during a temporary Presidential disability preside over the Senate and cast a deciding vote on his own veto? I presume the answer is no one knows for sure.
The VP only votes in the Senate when it is “evenly divided against itself.” Overriding a veto requires a 2/3 majority, so if the Senate were 50-50 to override the veto would be sustained whether the VP voted or not.