The patient's right is to find another doctor on their own accord...they're not locked into the conscientiously objecting doctor.
That's a very simplistic and somewhat unrealistic conclusion for you to come to.
It's just not that easy. As BA22 said, there is a power imbalance between the doctor and patient. Some doctors can be very convincing and powerful bullies. They can intimidate their patients from seeking a second opinion, especially if they have strong beliefs about what the right thing to do is.
Me and my parents have experienced this several times. Doctors have said stuff like:
-"I'm NEVER WRONG! I see that as a personal insult if you seek a second opinion. The hospital pays me a lot of money to do what I do." (A neurologist at the royal children's hospital)
-"If you go and see that doctor at that hospital, don't ever expect to step back inside this hospital again." (My cardiologist)
-One doctor rang up my dad and said, "How dare you even consider getting a second opinion. If you do, I will never see you as a patient again. And think about it; I can easily have an influence on what your reputation is as a doctor." (This was my dad's gastroenterologist)
So yeah. It can be really difficult to then seek a second opinion when you have essentially been threatened. You'd be surprised how often it happens. All my friends with cystic fibrosis have had very similar experiences. And, you've got to realise that it's difficult to seek a second opinion in the first place. It's hard to just go and find another doctor, especially when you are in such an awful situation.
When you are registered as a doctor, you agree to work by a code of practice. No one is being forced to work in an industry which requires you to refer patients if you are unable to treat or give advice on a particular problem yourself. If you don't like it, don't practice! And, Medicine is a career which requires you to be fairly level headed and balanced.