Unless this software is running only on the patient's own personal computer, and is not networked anywhere, such a system in the US would need to comply with the requirements of the Health Insurance Portability and Accountability Act (HIPPA). This requires that "protected Health Information" (PHI) be stored protected by encryption or other security methods approved by the federal government, that users of the system be restricted by enforceable contracts not to improperly access or use such information, and a number of other requirements. I work on software designed for hospitals which stores patient information, including Rx and diagnosis info, and we have to take this into account at every stage of design. The penalties for violating the HIPPA rules are sizable, potentially enough to bankrupt even a major corporation (although they usually are not enforced to the theoretical legal limit).
I do not believe that you need an advance license, but you need to review the HIPPA requirements very carefully.
Exactly what precautions such software would need to take would depend on how it is intended to be used, and by whom, and where PHI would be stored. This is a potentially tricky area, and you should consult both legal and software experts in the specific field of HIPPA compliance, before trying to market any such software, or load information about any real people into it. Test data about purely fictional individuals is safe.
If the software is to be used to trnasmit prescriptions for dispensing, there are additional state and federal requirements under the Controlled Substances act and various state laws. These may require a permit in advance, I am not sure.