ANSWER:
Yes, Paul did leave Trophimus sick in Miletus. What does that prove? Paul couldn’t heal everyone. But Jesus couldn’t heal everyone either.
Even though healing is provided in the atonement, not everyone is going to get healed. It’s not a defect in the atonement, nor is it an issue of God’s will, it’s an issue of faith. That’s the answer Jesus gave to the disciples when they asked him why they didn’t see the epileptic boy healed. The disciples obviously believed that they ought to, and should have been able to, heal him. They tried, but never concluded (as we wrongly do) that it wasn’t God’s will to heal him. Jesus told them squarely it was their faith issue. Jesus said NO HEALING will be impossible for you if you have the faith for it.
19 Then the disciples came to Jesus privately and said, “Why could we not cast it out?” 20 So Jesus said to them, “Because of your unbelief; for assuredly, I say to you, if you have faith as a mustard seed, you will say to this mountain, ‘Move from here to there,’ and it will move, and
nothing will be impossible for you
(Matthew 17:19–20).
But Jesus could not heal everyone in His hometown of Nazareth either, not because of His faith, not because of God’s will, but because of their faith issues.
5 Now He could do no mighty work there [Nazareth], except that He laid His hands on a few sick people and healed them. 6 And He marveled because of their unbelief
(Mark 6:5–6).
When someone is not healed when we pray for them, there is an issue. It’s not a God issue. That’s a done deal. It’s a faith issue and a lot of believers don’t want to come to grips with issues of their faith, or someone else’s faith. So, it’s easier to put the blame on God and make judgments about His will to heal. When Jesus could not heal in Nazareth, He did not conclude it wasn’t His Father’s will to heal them. When Jesus gave healing authority to His disciples, they didn’t come back and question Jesus’ will to heal, when they were unsuccessful. They asked Jesus, “Why could WE not heal?” Jesus gave us the ability, authority, and anointing to heal (Matt. 10:8). So, when we don’t see healing, why do we question God’s will? We should be questioning our faith in the ability, authority, and anointing that we have been given.
So, it should not shock us when we read of an occasion when Paul was not able to heal. We don’t know the situation with Trophimus. And we don’t know how long after Paul prayed for him that he recovered. All we know is that Paul did not wait around for his recovery.