In most places, no. Many employers are surprised to learn that they are not legally required to give their employees paid time off (although a few localities, such as San Francisco and the District of Columbia, as well as the state of Connecticut, require employers to offer some paid sick leave). Despite this legal leeway, however, most employers do offer some variety of paid leave. A generous leave policy can help you attract high-quality employees and improve office productivity and morale.
If you decide to adopt a policy that gives your employees paid vacation or sick time, be sure to apply the policy consistently to all of your employees. If some employees receive a more attractive package than others, you might be vulnerable to a discrimination claim.
For more information on paid time off, see Nolo's article Providing Vacation and Sick Leave.