Yes, it could.
1. Most hospitals are operating at a loss, thus do not have money to maintain a strong infrastructure. Meaning that website could be on the same pipe as the rest of the organization.
2. The web sites now offer more services to patients and other providers. Such as logging in and able to send their Medical Records, Book and Appointment, or securely send a message to your doctor.
3. If you kill the Router (That devices that will need to direct you to the website) as the device wouldn't expect that type of load, it causes problems with other places it is trying to work with. As the software will often need to talk to other locations. For example Lab work is often done at different locations then sent over to your Dr. for review. If the network goes down the Dr. may not get the results.
In short you do not know how things are connected, and could be combined with other things that you don't expect.
To say, It is only a website doesn't mean DDoS the site will not cause other problems.