[note: correction below]
The title of my question did not capture my original intent, which was to focus on holding both jobs at the same time. Since that intent was generally understood, I will award the "selected answer" accordingly.
But as for editing the title of the question, it does not feel fair to do so after so much interest and lively debate have been generated. This leads to a situation where, technically, the question as stated remains unanswered.
So for completeness:
Can the president appoint him- or herself to the Supreme Court?
No, because the President does not appoint anyone to the Supreme Court. He or she only submits a nomination which is subject to confirmation by the Senate.
And the obvious follow-up:
Can the president nominate him- or herself for the Supreme Court?
Yes. But judicial action or failure in confirmation might block the appointment.
From the comments, @phoog points out that, in fact, after Senate confirmation, it is the duty of the President to formally make the appointment. My mistake in the above was to use information from Wikipedia, which currently states:
Article Two... requires the President... to nominate Supreme Court Justices and, with Senate confirmation, requires Justices to be appointed.
The passive sentence ends there (without a "by" clause), so it puts focus on all of the "requiring" that "Article Two" is doing, without having to commit to who's doing the nominating (President), confirming (Senate), or appointing (President). Worse, I misread it, getting that the Article itself (or the Constitution) was authorizing the appointment; I actually remember thinking at the time that that was kind of weird.
Anyway... I should have simply checked the Constitution itself. But it can be tedious to read, due to its exuberant use of embedded clauses. So here's the the actual text of Article 2, Section 1.7, with the excess clauses excised to focus on the relevant issue. I also added a set of parentheses indicating my interpretation of the remaining clause structure:
..., and [the President] shall... nominate, and [,] (by and with the Advice and Consent of the Senate), shall appoint... Judges of the Supreme Court.
So now, the question that's raised is whether this designation of Chief Executive as "appointer" is simply a mysterious formality, versus it possibly introducing tangible or even substantial effects.