The article "Enforcement of Religious Courts' Judgments Under Israeli Law" (Asher Maoz, Journal of Church and State 1991) is useful in understanding the legal underpinnings of this question. The beginning point is the Palestine Order in Council 1922 (this is from the UK Privy Council), where religion-based courts are established, so sect. 52 is about Muslim courts, 53 (later repealed) is about Jewish courts and 54 is about Christian courts. Section 53 was replaced with Rabbinical Courts Jurisdiction (Marriage and Divorce) Law, 1953, stating that marriage between Jews follows Jewish law, and says that rabbinical court has exclusive jurisdiction. Article 6 of that law allows a district court enforce a final judgment by a rabbinical court by imprisonment. Muslim courts similarly have exclusive jurisdiction over Muslims, though it is wider (it pertains to all matters, not just marriage).
Actual enforcement apparently involves the Chief Execution Officer. Basic Law 15 gives the Supreme Court supervisory power over all courts, including religious courts. So rabbinical courts don't have unfettered power to do whatever they want (under Jewish law), but there is direct state sanction of imprisonment as a penalty for not obeying a rabbinical court decision, there is non-religious execution of court orders, and there is secular supervision. Since civil marriage is non-existent in Israel, there is no choice (if you want to have the wedding in Israel) but to go to the recognized religious authorities, which seems to make Cyprus a popular wedding venue. A Roman Catholic divorce is not possible in Israel, a Muslim one is, and I really cannot tell about the various Orthodox Christian churches.