Can God heal? Yes. Does God heal? Absolutely. Jesus healed many as a part of His earthly ministry. He still works miracles at times, some of which include miraculous physical healing. But Jesus didn’t heal all sick people. Why? Because it wasn’t His primary mission and purpose. He came to save the souls of sinners, not to merely make their present life more comfortable (Mark 2:17). Surely, He cared for their needs, hurts, ailments, and discomforts. But He did not heal every physical ailment present on the earth. In fact, though there is nothing recorded in Scripture to this effect, it is possible that Jesus Himself was sick or injured at one time or another (besides the time leading up and including the cross, of course). Sickness is not necessarily a result of sin, but it is part of being in a cursed world and in a mortal body, as Jesus was (Philippians 2:6-8, Hebrews 2:14).
Sometimes we might be prone to think that Jesus would never have us be sick or hurting. The reality is that one day He will wipe away our tears and remove all pain and grief (Revelation 21:4), but that day has not yet come. There is a present purpose for suffering as it shapes us, challenges us, and draws our hope toward eternity (James 1:2-4, 2 Corinthians 4:16-18). But there is nothing in Scripture that guarantees that in the cross all sickness has been conquered. How do we know this? Well, the Bible is explicitly clear about this. Sometimes we get the idea that the apostles were healing everybody whom they came into contact with. We might tend to think that they had a sort of indiscriminate healing power and ability. We might even be tempted to think that they could tell God when and where they wanted Him to work and heal through them. The truth is that the apostles could only do what God would enable them to do by faith. When they moved to heal someone, they were moved by the Spirit, confident that they were to believe God for healing. But there were other times when God didn’t heal, and they weren’t in shock or awe about this. It didn’t weaken their faith in any way. In 2 Timothy 4:20, Paul says, "Erastus remained at Corinth, but Trophimus I left sick at Miletus." Paul, the apostle who himself was healed from a deadly bite by a poisonous viper (Acts 28:3-6) could not or did not heal Trophimus. He left him sick. Why would he do this if he had the power in Jesus to heal him? Certainly, both of them would have believed for healing and then gone on to serve the Lord. But God chose not to heal, and Paul left Trophimus sick. What about Epaphroditus, a helper to Paul and servant to the Philippians? He was sick to the point of death, but the Lord spared his life (Philippians 2:25-30). Why did things have to get so bad to the extent that this righteous servant of God almost died? Couldn’t Paul or a person "gifted in healing" have spared him this near death experience? No, because man cannot manipulate God, and God heals when God wants to heal. What about Timothy? Paul told him in 1 Timothy 5:23, "No longer drink water exclusively, but use a little wine for the sake of your stomach and your frequent ailments." For whatever reason, Timothy had some stomach issue and other frequent ailments, perhaps related, perhaps not. Regardless, he was regularly suffering from some variety of sickness. Paul’s advice was not to exorcise some demon of stomach sickness, nor did he come to Timothy and bestow some healing power upon him. Timothy was a man of great faith and was one of Paul’s closest friends and fellow servants of the Lord. It wasn’t for lack of faith that he wasn’t healed. It just wasn’t God’s plan for some reason, and both Paul and Timothy accepted that and lived in light of that reality. They took practical steps to deal with the problem. There is room in the Bible for doctors and medicine. Luke, after all, the author of the gospel of Luke and the book of Acts, was the "beloved physician" (Colossians 4:14).
I selfishly wish that we could always be healed because of Christ’s victory over sin and death on the cross, but the reality is that even in Jesus’ time and in Paul’s time and personal life, healing wasn’t always a reality. When God wanted to do something special and significant, He healed. At other times, sickness was a trial or a way of life. In these cases, as in all instances, God’s grace was sufficient (2 Corinthians 12:9-10).
We must not presume upon God’s healing, though we ought to believe in it. God does heal (James 5:13-16) even today, but it is because He wants to. He honors the prayers of faith from the righteous hearts of His children, but He doesn’t always work how we want Him to work. I have been amongst believers that have firmly believed God for healing, and for whatever reason, God has chosen not to heal. Sometimes, people even go on to be with the Lord. Yet I have also heard of many testimonies where God has confounded doctors and miraculously healed. Regardless of what God decides to do, we can trust that His way and will is best. We cannot command God to heal, but we can ask Him to. We can believe Him to, but we cannot require Him to. In this world, we will have tribulation, but we ought not to fear because He has overcome this world. God causes all things to work for our good, even sickness, and, yes, even death. As Paul said, "To live is Christ, and to die is gain" (Philippians 1:21). One day sickness won’t be an issue anymore, but until we are with Jesus, let us pray for healing and grace to endure all things.