So if I were to see a doctor, would that mean my heart isn't right with God?
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The Word says to go to God first, or He is not your Lord. When you were a child and something was up, did you say, well, I better go see my doctor and figure this out, OR did you go to Daddy and Mommy and tell them about it (not holding anything back or expecting to have to go anywhere else)? Now we're called God's CHILDREN, not His adults, so this also would indicate that as a much-loved child of the Most-High God, that He wants us going to Him first, with no secondary objective in mind, or expecting whatever God would do for you to fail or expecting that He either cannot or will not do it.
A backup plan means you expect the first to fail (it's being double-minded), and accordingly, as James puts it, the double-minded man shall expect to receive absolutely nothing of the Lord.
You can be saved and still not believe God for healing. It doesn't make you less saved or inferior to anyone else, but you can still miss out on the Covenant promises. See Jeremiah 17:5-8 to see that in action - both are Covenant people, both live in the same area, but only one receives what God has given through the Covenant, the other makes man his arm (flesh his arm; in other words, he looks to man to get his needs met).
With all that said, if you sincerely went to God as His child and called on Him to do as the Covenant promises and you resist that sickness or disease or problem, having bound the Devil already over it and you confess what God says in the Scriptures concerning you and He answers by telling you to go to the doctors, then do it.
(Though God already supplied you with the example through Jesus, showing that whoever believed He was the Christ, sent to them to heal them standing in the place of God, receiving Him as sent to them BY God to do what they need and want, He healed them ALL and never sent one to a doctor - I point it out, because HE said, in the Holy Bible, ALL things are possible to him who believes (that's why you see Him have a blind man stand before Him and ask him what he wanted Him to do, and then after the guy says I want to see, Jesus further asks, "Do you believe I can do this?"....now keep in mind, this man did not know anything about Jesus except He had heard He was David's son, the Messiah). He was persistent and did not give up, not caring what anyone thought - he knew this was his chance to receive the touch of the Messiah and SEE! And he wouldn't quit, his faith was in his heart AND in his mouth.