— Махатма Ганди pre-eminent leader of Indian nationalism during British-ruled India 1869 - 1948
Widely attributed to Gandhi, sometimes citing Ramachandra Krishna Prabhu, The Moral Basis of Vegetarianism (1959). (Cf. Ralph Keyes, The Quote Verifier (2006), p. 74.) However, it is not found in that essay http://www.mkgandhi.org/ebks/moralbasis_vegetarianism.pdf nor in any of Gandhi's Complete works. http://animalsmattertogod.com/2013/09/13/mahatma-gandhi-hoax-quote-greatness-of-a-nation-and-its-moral-progress-can-be-judged-by-the-way-that-its-animals-are-treated/
The original quote seems to be by David Strauss, The Old Faith and the New (Der alte und der neue Glaube, 1872, trans. by M. Blind, New York: Henry Holt & Co., 1873), vol. II, ch. 71, p. 59 https://archive.org/stream/oldfaithnewconfe01stra#page/59/mode/2up: The manner in which a nation in the aggregate treats animals, is one chief measure of its real civilization.
Attributed to Gandhi since at least 1983: Gandhi is quoted as saying, "The greatness of a nation and its moral progress can be judged by the way its animals are treated." Description of the film Vegetarian World in Vegetarian Times, issue of July 1983, p. 26 https://books.google.com/books?id=7QcAAAAAMBAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q=%22gandhi%20is%20quoted%20as%20saying%22&f=false. In this source, the quote is claimed to be taken from the William Shatner documentary The Vegetarian World (1982). A recording of that film is available on YouTube https://www.youtube.com/watch?v=adZnCsVYnW4: although the documentary does mention Gandhi, it does not appear to attribute this or any similar quote to him.