An Islamic tribunal, launched in Dallas last year to settle civil disputes among the growing Muslim population, is drawing quarrels over its use of Shariah law. Opponents fear it would open the door to extreme practices and corporal punishment linked to Shariah law in certain Muslim-majority countries. Still others argue it would replace the U.S. Constitution. But organizers say the panel of arbitrators issues nonbinding decisions on matters such as business disputes and religious divorces. They note its parallels with Jewish rabbinical courts and Catholic tribunals. The very word “Shariah” invokes backlash from those who don’t understand it and use...