Yes and no. I screwed my shoulder up in an accident 2 years, 1 month, and 3 days ago ... It definitely doesn't hurt as bad now as it did back then, even after surgery...but there are days when it really keeps my attention. If your surgery was really recent, then yes, it is normal to be in serious pain. Follow dr orders, and take it easy. It will eventually start to feel better. I wish you a speedy recovery!
It hurts now because your shoulder is still healing from the surgery. Eventually the pain will go away, though depending on what type of surgery you had there may be things that make it hurt - lifting too much weight, cold weather, turning it the wrong way, etc.
You will heal. Word of advice, as a massage therapist I work on a lot of rotator cuff injuries/surgeries. Do Everything Your Physical Therapist Tells You To Do. The patients I work with who have done all their prescribed exercises have regained full range of motion, those that didn't do the exercises ended up really limited, though none of them had lingering pain.