Employment Law Questions? Ask an Employment Lawyer.
Yes, they can. HIPAA doesn't stop someone for asking for information. What is does is stop the holder of information from giving it out without your permission.
An employer is not required to give sick leave. When they choose to, they can create reasonable limitations to its use. If the employer only allows sick leave in a specific way, with specific evidence, they can legally request the information they are asking for. You have the legal right to refuse, but then they don't have to grant the request for sick leave.
However, the other issue is that you missed time. If they wish to consider the time unexcused then, based on your not giving them the medical proof, then it can legally be counted against you.