No Federal law and no law in any state requires an employer to provide EITHER vacation or sick time for any employee. Both are entirely at the opt of the employer.
That being said, while the law does not require an employer to provide any paid time off benefits, there are laws that affect the situation you describe, but we don't have enough information to say if they apply. So this is what we need to know:
1.) Are you exempt or non-exempt? Salaried is only a pay method and has no legal status of its own
2.) Does your employer offer paid sick days at all?
3.) If so, how many?
4.) Had you used any previously?
5.) Did you miss any additional time other than the half day?

