I subscribe to the view that the reason is because the gifts that he was performing in the early days was a "sign to the Jews"; not all spiritual gifts are still in operation today; or, they aren't necessarily "needed." The gift of tongues and the gift of healing has ceased. The keyword here is "gift." This isn't to say that God doesn't or won't heal today, because this is obviously not true, but the "laying on of hands" by a man (as a 'gift') has all but ceased, and yet there are still 'healing teachers' out there in the world today teaching a false gospel, having "healing revivals" that come to nought, etc. (Many of these 'teachers' are a part of the 'Word of Faith' movement, the Properity Gospel crowd, et al, that teach that positive confession is necessary for having what you want or "claim," and they teach that if you do not get healed, or if you get sick or have any sort of malady, this means that you don't "have enough faith." Many in this movement even say shocking claims such as the erroneous idea that being poor is a sin, and that if you are sick, "it's your fault that you're sick." This is Heresy 101 at its best!)
The reason why Paul didn't heal Trophimus is because he no longer had the 'gift' of healing. Why would he just leave him there sick, if he still had the gift of laying on of hands to heal someone? The answer is: He wouldn't have!
The gift of healing by the laying on of hands, etc., was no longer needed, because that particular gift was a gift for a sign to the Jews! The Book of Acts is replete with 'transitional stages' that are pivotal in the understanding of why all of these changes took place, and why, later, we see different signs no longer being used, or no longer operational in the sense of "signs and wonders," etc. There no longer needed to be "signs and wonders" later on when Paul revealed to them that he had received the "revelation" from Jesus Christ on what they all were to go on preaching from them on, which is "the gospel" according to 1 Corinthians 15:1-4 (which Peter and the others agreed to in Acts 15 at the Council of Jerusalem).
If the Apostle Paul still retained the gift of healing by laying on of hands (or healing, in general), then there is absolutely no way under the sun that he (Paul) would have left any brother sick. In fact, in 1 Timothy 5:23, look what Paul recommended:
Drink no longer water, but use a little wine for thy stomach's sake
and thine often infirmities.
In other words, he is suggesting to use "medicine." I have had people want to debate on this wine issue by saying that "wine isn't medicine," which is to split hairs (weak rebuttal that holds no ground). The context is that Paul was suggesting for this brother to use wine to help him with his stomach sickness. If the gift of healing was still in operation, it's unlikely that Paul would have said to use something to merely "help" with your "infirmities." In fact, he would have probably sent a handkerchief or even went to him, personally, if possible, and laid hands on him and healed him instantly. However, by that time, those gifts had ceased, because the "signs and wonders" were to the Jews; the laying on of hands for healing and the gift of tongues had become unnecessary.
The Apostolic Gifts are no longer in operation today: 1) Laying on of hands to heal someone (if this were still the case, then people could walk into nursing homes and hospitals and begin healing people left-and-right, but this doesn't happen. God heals when it's in His will to heal; the difference is that the gift of healing (a man laying on of hands with instant healing) has ceased, and 2) so has the gift of tongues (it's no longer needed, but if it is used, I believe that God will allow it to occur, only when the church is edified, and only when there is an interpreter amidst the assembly! Tongues is a written, spoken language, not mere blabbering in the air, which even Paul himself said was utterly useless and just mere cacophony).