There is nothing in the Bible that says we cannot use doctors. In fact, Luke is commonly believed to have been a physician. So, God has chosen a doctor to help write the New Testament.
God has gifted believers, as well as unbelievers, with the ability to study medicine and to be used by Him in order to heal people. In fact, God has gifted all kinds of people in all kinds of ways. What about going to the orthodontist to have our teeth straightened?
As Christians, should we just pray about it and not go? Or what about people with bad vision? Should they pray and ask for healing or go to the ophthalmologist? The answer is simple. Do both. Pray and ask the Lord to heal you. If He doesn't, go to a doctor.