Did the Islamic Sharia descend to be ruled by Muslims only, or to be applied to followers of other religions as well?

If the People of the Book (Jews and Christians) live in Muslim lands as Dhimmis (protected people) or Musta'minun (those granted safe conduct), they are allowed to practice their religion and live under the authority of Islamic Law. A Muslim judge must rule among them according to Islamic Law if they refer their disputes to him, while observing the jurisprudential rules that state, "We are commanded to leave them and what they believe in" concerning their private beliefs and rituals in their homes and places of worship. However, if they openly display anything that contradicts Islam, or if the dispute involves civil or criminal matters, then Islamic Law applies to them. The Prophet's (peace be upon him) treaty with the Jews stipulated recourse to him in disputes, which refutes the argument of those who reject the application of Islamic Law to non-Muslims. They do not have the right to refuse the rulings of Islamic Law if they accept man-made laws that have no relation to their own religious codes.