Can God use a non-believing doctor to heal me?
A: Yes. God is so powerful that he uses atheists and people of other religions every minute of every day to accomplish his purposes. Those being used don’t know that they are agents of God’s special grace, but that doesn’t stop God.
It is really nice when you can find a medical professional who is a believer and who prays but don’t fret if you cannot. Some have chosen less-skilled doctors on the basis that they are Christians. This is a personal preference, but I don’t think it is necessary to do that. God is sovereign and in control of all and can use anyone for his purposes. A non-believing doctor can have God’s blessing because of the prayers of believers for him or her in your situation.
Whether you are being treated by a Christian or a non-Christian doctor, a Christian should always pray first, during, and after treatment for miraculous healing and never give up on the reality that God may yet heal miraculously. I believe God has an especially good time waiting until someone is in treatment to come through and heal them because he likes to show hardened medical professionals that he is the greatest physician that there is.