How do we know that the Sabbath is in fact on Saturday and not on Sunday or Friday as observed by Christianity or Islam?
Interestingly, the Talmud (Sanhedrin 65b) raises your question. The distinguished military leader Turnus Rufus asked Rabbi Akiva what makes the Sabbath special. Rabbi Akiva replied, “What makes you special?” He answered, “The king desired to elevate me above all others.” Rabbi Akiva said, “So too G‑d elevated the Sabbath above all others.” Turnus Rufus challenged, “Who said today is the Sabbath? Rabbi Akiva replied, the River Sambatyon [which miraculously rests on the Sabbath, see The Lost Tribes 1 – Where Are They?] is proof, Ov [a form of sorcery which doesn’t work on the Sabbath] is proof and your father’s grave that doesn’t raise smoke [since the wicked are given respite] on the Sabbath is proof.”
The seven-day week has always been the universal, standard method of counting the weeks, and the seventh day of the week has always been observed by Jews as the Sabbath: “Work shall be done on six days [corresponding to the days of Creation], and on the seventh day you shall rest.” In Hebrew, the weekdays are named ‘day one’ through ‘day six’, and the Sabbath is called Shabbat, the day of rest. Even the religions you mention agree about which day is the Sabbath: the Latin-based languages refer to the day with some variation of Sabbath, and in Arabic the weekdays are named by number as in Hebrew, while Saturday is called ‘yaum as-Sabt’. If so, why don’t those religions observe Shabbat as their day of rest?
Shabbat is Divinely ordained for the Jews alone. The verse states, “And the Children of Israel shall keep the Sabbath as an eternal covenant between Me and [them]” (Ex. 31:16). Our Sages taught (B.R. 11:8), Shabbat complained before the Holy One saying, “Every day of the week has a mate: day one has two, three has four, and five has six. Who shall be my mate?” G‑d replied, “The Jewish people are also alone among the nations. Israel shall be your mate!” Similarly, the Shabbat prayer proclaims, “You did not give [Shabbat] to the nations of the lands…but to Israel Your people have You given it in Love, to the seed of Jacob, whom You have chosen”. Therefore, despite the fact that these religions were inspired to the idea of the Sabbath through Judaism, by divine intervention their observance was deflected away from Shabbat to Sunday or Friday.
Our sources note that according to ancient Astrology (which Judaism recognizes, see The Zodiac), Saturn’s influence over Saturday dictates that the day should be one of sorrow and pain. For this reason, the early Christians, who were knowledgeable astrologers (note that the Latin names of the days of the week correspond to the names of the luminaries which accordingly govern the beginning of each day), fixed Saturday as a day of solemn fasting and self-affliction. Perhaps under this pretense they were divinely deflected away from Shabbat (R. Yonaton Eibeshitz, Ya’arot Dvash 2:3).
However, the Talmud (Shabbat 156) asserts that the Jews are not bound by astrological influences. For this reason G‑d commanded to observe Shabbat not only by resting and refraining from work, but also by actively proclaiming and sanctifying it through kiddush, meals, singing and learning. In this way G‑d reveals the unique, eternal and unchanging relationship between “Me and the Children of Israel” (R. Bachaye).