I take my service dogs to all my medical appointments including blood draws, emergency rooms, dentists, hospitals, and even surgery!
Under the ADA, they are allowed in non-sterile areas. If you are allowed to wear your street shoes, it isn’t sterile! When doing things like getting stitches, they do create a sterile field– this is a small area that is sterile, and does not preclude your service dog from being in the room. They can deny access from the actual surgical suite, but not from pre- and post-op recovery rooms.
When I’ve had surgery, at my pre-surgery appointment I bring information about the ADA laws and from the Centers for Disease Control just in case they have questions. I explain how I need my dog before and after the surgery (making sure to emphasize how much less work I will be for the nurses with my dog to help me!), and that during the surgery my husband will hold my service dog. They usually allow him to bring my dog back to recovery when I start to wake up.
During some procedures that cause me anxiety (like dental exams), I have my service dog lie on top of me to provide pressure therapy. This is possible even with a large breed service dog, though it is easier with a small one!
Below are pictures and a video of me with my service dogs in various medical settings over the years. Please leave any questions you might have in the comments!