Holy shit! There are some US Presidents- not to mention some current political figures- whom I would have no trouble believing held such nauseating views. But that FDR- whom it can be said without exaggeration is regarded as one of the greatest if not the greatest liberal icon of all American history- felt this way.... Jesus. At the risk of sounding melodramatic, I am never going to regard FDR in quite the same way ever again.
I guess then FDR would have recognized Israel- if he did @ all- only after a LONG delay.
As I already noted, he tried, without success of course, to persuade King Ibn Saud that letting more Jews into Palestine would be good for the Arabs as well as the Jews, so he was evidently not lacking in some sympathy for Zionism.
As for what he would do in 1947-8, like Truman in OTL he would probably be torn two ways--between a national security establishment worried about alienating the oil-rich Arabs and his own sympathy for Zionism and realization that the Jewish vote would be important in 1948 (I don't think he'd run himself but he would want a pro-New Deal successor to win). One must remember that Truman himself was not nearly pro-Zionist enough for many American Jews; he only recognized Israel de facto, not de jure, and he imposed an arms embargo that was widely seen as hurting Israel more than the Arabs. This probably cost him New York, where Henry Wallace got a half million votes that FDR had received in 1944. Many of them were from Jews, and while some of the Jewish support for Wallace was from leftists who would have backed him anyway, Samuel Lubell has argued that that the Palestine issue probably did motivate some ordinarily non-leftist Jews to vote for Wallace. It was fortunate for Wallace that the USSR was temporarily pro-Zionist, enabling the Progressive party to take a very pro-Israel stand. (Ironically, he privately supported the arms embargo!) Truman, btw, like FDR could make anti-Semitic remarks in private, once saying that it was no wonder he couldn't please the Jews since Jesus Christ couldn't do so...
BTW, Felix Frankfurter, who continued to advise FDR on political and legal matters after being appointed to the Court (that was commonplace at the time though it would be considered a breach of judicial ethics nowadays) would also be a strong voice for Zionism.
As for occasional anti-Semitic remarks by FDR, there is absolutely no reason to be shocked by them; they were commonplace at the time--including the belief that the Jews were to some extent responsible for gentile hostility toward them. Lloyd George said at Versailles that "There is obviously something to be said to justify the hostile feeling of the Poles against the Jews. M. Paderewski told me that, during the war, the Jews of Poland were by turns for the Germans, for the Russians, for the Austrians, but very little for Poland herself." (Wilson replied that "It is the result of long persecution. The Jews of the United States are good citizens...Our wish is to bring them back everywhere under the terms of the law of the land.")
https://books.google.com/books?id=rRrRBgAAQBAJ&pg=PT146
As for FDR's belief that scattering Jews around the world would help diminish anti-Semitism by avoiding too large a Jewish concentration in particular countries, this at least did recognize that the Jews of Europe did have to go
somewhere if they were to survive. And FDR did actually do something, though of course too little, on this belief: "Over protests from his notoriously anti-Semitic State Department, moreover, Roosevelt encouraged efforts to settle European Jews in Latin America — about 40,000 of them made it there from 1938 to 1941 — and pressed the British to keep Palestine open to Jewish refugees."
https://www.nytimes.com/2013/04/07/...by-richard-breitman-and-allan-j-lichtman.html Of course he should have pressed for the admission of more Jews into the United States, but it is not clear that he would have succeeded--Congress was notoriously hostile to immigration during the Depression.