Is an employer required to give employees paid sick days?

The short answer is no. The longer answer goes something like this:
- Non-Exempt employees never have an inherent right to paid sick time, or paid vacation, holiday, jury or any other type of paid time off. The employer can maybe create a policy or contract that requires something but there is no externally imposed requirement. Any one paid overtime is probably legally Non-Exempt.
- Most Exempt employees are paid on a Salaried basis and Exempt Salaried employees only have certain restrictions on having their salary docked. These restrictions can include sick days, although this is one of those "read the fine print" kind of restrictions. Exempt employees paid on other then a Salaried basis do not have this restriction.
http://www.dol.gov/dol/allcfr/ESA/Title_29/Part_541/29CFR541.602.htm

These are federal rules. San Francisco has a city ordinance currently tied up the courts that mandates paid sick leave be offered, but the press reporting is pretty much that the world will end if the ordinance is upheld. If there are states with this type of requirement, it is being kept quiet. I am sure that there are no federal requirements other then what I mentioned.