Q:

Why didn't Einstein win a second Nobel Prize, for relativity?

Einstein's failure to win a Nobel until 1921, and that prize's not being awarded for his work on relativity, is generally ascribed to these factors:

Lack of sufficient experimental proof for the theories at the time (some distrust in the early results on the precession of Mercury, and the Eddington starlight observations).
Failure by some members of the committee to understand relativity
anti-Semitism
skepticism generally about the utility of relativity, and whether it was physics at all.

If these factors were in fact the reason that the relativity work did not garner a prize in 1921, why didn't the relativity work merit a second Nobel prize after, say 1945? Surely the work on cyclotrons and the bomb provided firm experimental support for the validity of the theory, and by then the overall utility would have been understood as well.
(For that matter it's interesting to speculate how many Nobel Prizes Einstein should have won; but this question just asks: why only one?)

A:

The time when experimental evidence supporting relativity began to appear coincided with the rise of antisemitism in Germany following Germany's defeat in WWI.

Antisemitism was on the rise in Germany; Jews were being scapegoated for the country's defeat in the war. As both Jew and pacifist, Einstein was an obvious target. The complexity of relativity did not help either. Opponents such as Ernst Gehrcke and Philipp Lenard found it easy to cast doubt upon its labyrinthine mathematics.
The situation reached crisis point in 1921 when, paralysed by indecision, the Nobel Committee decided it was better not to award a prize at all than to give it to relativity. The arguments raged for another year until a compromise was reached.
At the suggestion of Carl Wilhelm Oseen, Einstein would receive the deferred 1921 prize, but not for relativity. He would be given it for his explanation of the photoelectric effect, a phenomenon in which electrons are emitted from a metal sheet only under certain illuminations. The work had been published back in 1905.

Thus, after considerable hand-wringing and even deferring the prize for a year, in 1922 the Swedish Academy made the decision to award Einstein the 1921 prize a year late, writing to Einstein on the 10th of November 1922 :

... the Royal Academy of Sciences has decided to award you last year's Nobel Prize for physics, in consideration of your work in theoretical physics and in particular your discovery of the law of the photoelectric effect, but without taking into account the value which will be accorded your relativity and gravitation theories after these are confirmed in the future.

(Source : Mass and Energy in Relativity Theory, by Lev Borisovi.)
At this stage relativity was open to future consideration. However, Einstein chose not to attend the ceremony to receive the award in person. This decision may have made a future prize for relativity more difficult, if not impossible, since it gave the impression that Einstein did not value the award.
Having said that, at the time Einstein received confirmation of the award from the Nobel committee, he was committed to a lecture tour of Japan.

German foreign minister Walther Rathenau had been murdered by anti-Semites. In the subsequent investigation, the police had found Einstein's name on a list of targets. In the face of such a death threat, leaving Germany to spend months in the Far East, rather than a few days in Stockholm, must have seemed prudent.

A:

A short answer to the question might be that, after the Nobel award to Einstein in 1922, many people were given the impression that it had been awarded for relativity, and that consequently he received few or no further nominations for any second Nobel prize. I recall seeing a biographical statement to this effect (but can't now locate the reference).
What does emerge from citable records, however, and makes any shortage of further nominations understandable, is that much in the record could well give rise to a mistaken idea that Einstein's prize actually had been awarded to him for the theory of relativity.  The following examples would contribute to such a mistaken impression. 
First, there was confusion in the media about the subject of the award.  When the New York Times  announced the award to Einstein, it ambiguously mentioned relativity, but not a word about the actual subject of the award, Einstein's study and results about the photoelectric effect: 

"NOBEL PRIZE FOR EINSTEIN. The Nobel committee has awarded the physics
  prize for 1921 to Professor Dr. Albert Einstein of Germany, identified
  with the theory of relativity, and that for 1922 to Professor Neils
  {sic} Bohr of Copenhagen." 

-- from New York Times, Nov. 10, 1922, page 4.  
Second, confusion also arises from the circumstances of the award and the lecture that followed it. The usual pattern is that Nobel laureates give a lecture about the subject of their prize.  There is indeed a lecture by Einstein in the Nobel archives, but it is not about the subject of his prize, it is about relativity: The Nobel Prize in Physics 1921 -- Einstein Nobel lecture (July 11, 1923).  The background explains this to some extent. When Einstein received news of his award, he was already on his way to Japan, and perhaps could not attend the award ceremonies in person, at all events he did not attend (Abraham Pais (1982, Oxford UP), 'Subtle is the Lord' : the science and the life of Albert Einstein' at p.503).  At the award ceremony on December 10, 1922, Rudolf Nadolny, German ambassador to Sweden, accepted the Nobel prize in Einstein's name and made diplomatic acknowledgements on his behalf -- and naturally there was no scientific content in what was said. 
After the Nobel ceremonies, Einstein was invited to Sweden to lecture on a subject 'of his choice'.  Arguably this muddied the waters and made confusion worse about the subject of the prize. 

"In March 1923 Svante Arrhenius, one of the Committee members, wrote
  to Einstein suggesting that the latter not wait until December for his
  visit to Sweden but that he come in July. He could then attend a
  meeting of the Scandinavian Society of Science in Goteborg on the
  occasion of the 300th anniversary of the founding of that city.
  Arrhenius left to Einstein the choice of topic for a general lecture,
  'but it is certain that one would be most grateful for a lecture about
  your relativity theory'.  Einstein replied that he was agreeable to
  this suggestion, though he would have preferred to speak on unified
  field theory."

-- Pais (1982, cited above), p.505.  
Einstein, when he finally arrived in Sweden several months after the Nobel award ceremonies, did indeed lecture about relativity. His lecture was not formally linked to the Nobel award, it was given at a scientific meeting in Gothenburg.  But it is preserved in the archives among the Nobel lectures, and it is a little difficult to find indications that make clear the relation between the lecture and the prize. There is (now) a fine-print footnote added to the English translation of Einstein's lecture (but not with the original German version). The material is headed 'Nobel lecture' and the footnote to the English version runs: "The Lecture was not delivered on the occasion of the Nobel Prize award, and did not, therefore, concern the discovery of the photoelectric effect."
This combination of events and records was inherently likely to reinforce any confused idea that actually Einstein had already received a Nobel prize for relativity.  Mistakes and myths have arisen from lesser causes than that. 

