Well, as another person pointed out, what if the doctor was a Jehovah's Witness who disapproved of blood transfusions, while you need one. Or what if your doctor was a strict Jew or Muslim who refused to do work on the Sabbath, and therefore refused to help you until the next day?
So, your position that a Christian should not go to a non-Christian doctor is, in most cases, simple bigotry.
No, its a matter of associating with people who share your values and giving them your business, rather than supporting those who are undermining the type of society you wish to see come to fruition.
Such as a Jewish or Hindu doctor?