Birth certificates, and other vital records, are maintained and controlled by each respective state government. Some states are "open records" state, while other states have either no or limited access to particular records except for the subject or his/her next-of-kin. For example, Ohio is an open records state - virtually all vital records can be accessed by the public. VA, OTOH is not an open records state. To access some vital records, you must be next-of-kin, and you must provide identification, which will be photocopied and documented.
Hospitals are prohibited by federal law from granting access to any health records they maintain on current or prior patients, including the hospital's own copies of birth records. Most hospitals maintain copies of the birth certificates they generate for some period of time, usually decades or longer. Those particular copies are what are protected by HIPAA.
There are commercial sites available that do offer online access to birth documents. But, not every state participates, and not every birth document is available in every state. Again, it depends entirely on the state.
What is available in every state, is (generally) the name, the birth-date, and the sex of everyone born in that particular state and county. Sometimes this information is available online, other times you must appear in person.