If you read Rabbi Yosef's teshuva on saying Hallel (Yabia Omer v. 6 Orach Chaim siman 41 section 5) you will see that he bases his view of the Zionist State on an announcement published in 1949 before the first elections of the Knesset, signed by rabbis, urging Jews to vote for the United Religious Front. Rabbi Yosef writes that he read about this announcement in Rabbi M. Kasher's book Hatekufah Hagedolah. There Rabbi Kasher states that over two hundred rabbis from all over Eretz Yisroel, from all circles, signed on the announcement, which began with the words, "We thank Hashem that we have been privileged, through His great mercy and kindness, to see the first buds of the beginning of redemption, with the establishment of the State of Israel."

But the historian Tzvi Weinman later showed Kasher's claims to be a falsification of the facts. In an article published in Digleinu (Shvat 5739) during Kasher's lifetime, and later in his book Mikatowitz Ad Hei B'Iyar (pp. 134-136), Weinman showed that the original declaration read not "beginning of redemption" but "beginning of the ingathering of the exiles". Furthermore, the declaration was sent to the rabbis to sign, together with a notice that if the rabbi did not reply in the negative, his silence would be taken as consent to the declaration. (This explains why the signature of Rabbi Menachem Kuperstock appears despite that fact that he passed away 2 years earlier.) Furthermore, the declaration was sent in three different versions to different rabbis, and many of the rabbis signed on versions that did not include any positive words about the state, just about voting for the United Front. The 1949 activists who posted the announcement in the streets took all the signatures and put them under one declaration. Furthermore, there were actually two other announcement published at the time of that election, one signed by roshei yeshivos and one signed by Chassidic rebbes. Kasher took all the names of the signatories on all three announcements and claimed that they had all signed the one calling the state "the beginning of redemption".

Other than this evidence from Kasher's book, Rabbi Yosef, quite uncharacteristically for someone with his breadth of knowledge, does not give any Talmudic arguments or proofs to explain why it is permitted in his opinion to found a state.