In present day world, most of the democratic countries follow Secularism, i.e. separating state and religion. What is the opinion of Judaism on Secularism: Should a state following Halacha endorse Secularism? Should such a state keep religious matters separate from government?
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First of all, it cannot be emphasized enough that Judaism does not, and certainly since the days of the Sanhedrin, has not had, an official spokesman. Accordingly, it is essentially impossible to answer what Judaism thinks about this; only what individual Jewish scholars, basing themselves off, among other things, Jewish teachings, have held.
That said, below are a couple of rabbbis' views on the relative merits of a secular democratic state, in which religious institutions are not granted authority over the masses, vs. a theocratic state.
Here is what R. Shakh wrote about democracy (Mikhtavim u-Ma’amarim, vol. 5, p. 124):
On p. 127 he writes:
here is a passage, from R. Yissachar Meir, that appeared in an official Degel ha-Torah publication, Ve-Zarah ha-Shemesh (Bnei Brak, 1990), p. 630:
R. Elhanan Wasserman writes in his Ikveta di-Meshiha, par. 2, published on the eve of the Holocaust.
All of the above quotes are found here. For a discussion of rabbis who were supportive of communism, see here. This is significant since a communist state would be equipped to impose religious (or any other standards) on its citizens. This also demonstrates the degree of possible divergence of views within "Judaism".
Although rabbis Shakh and Wasserman, et al were opposed to Secularism, R. Schlesinger decided to set up his own system. This system he outlined his book Kollel HaIvirim. He explains his system would be democratic. He explains that the Torah requires one to follow the majority. This is so, even when a Godol or the like holds different views. He proves this by pointing to the system of the Sanhedrin. There, they did not just go with greatest Rabbi on the Sanhedrin, rather, they started polling the views of the lowest one. (Source)
This system seems much more democratic, and thus closer to secularism.
Additionally, R. Hayyim Hirschenson had positive feelings about democracy.