Islamic Law and Respondeat Superior
The doctrine of respondeat superior is well established in the legal systems of the United States and most Western countries.  As you have already read, under this doctrine employers can be held liable for the acts of their employees.  The doctrine of respondeat superior is not universal, however.  Most Middle Eastern countries, for example, do not follow this doctrine.
Codification of Islamic Law
Islamic law, as codified in the sharia, holds to a strict belief that responsibility for human actions lies with the individual and cannot be vicariously extended to others.  This belief and other concepts of Islamic law are based on the writings of Muhammad, the seventh-century prophet whose revelations form the basis of the Islamic religion and, by extension, the sharia. Muhammads prophecies are documented in the Koran (Quran), which is the principal source of the sharia.
An Exception under Islamic Law
Islamic law does allow for an employer to be responsible for an employees actions when the actions result from a direct order given by the employer to the employee.  This principle also applies to contractual obligations.  Note that the employer is responsible only if direct orders were given.  Otherwise stated, unless an employee is obeying a direct order of the employer, liability for the employees actions does not extend to the employer.
Question Presented
How would U.S. society be affected if employers could not be held vicariously liable for their employees torts?