Should I apologize?
No. You should maintain the feeling that the app is a neutral piece of technology, and the circumstances that led to the person being ineligible have nothing to do with the app. Some algorithm created a fact based on some factors, and that fact is being revealed. You should not be cruel or harsh with your tone when you explain this fact, but you should also not apologize.
Should I try to explain the rationale behind?
Yes. As much as is possible. This is your best opportunity to soften the blow. Even if it is just to explain that many complicated factors were used to make the decision. The worst thing is that the user feels like the decision was random or arbitrary.
You might consider sending unsuccessful applicants to a webpage that explains the process is slightly more detail (see last point, below)
Can I use visual elements or specific colors to improve this experience?
Again, this app is a neutral piece of technology, not a nurse or a babysitter. Design the screen that shows this message as you would any other screen. As soon as you start introducing additional visual elements, like images or color, you open the door for misinterpretation and a potential worsening of the situation.
Should I leave him with this last screen or propose some additional actions?
Only if there is a good reason to do so. Don't create UI that has to built, tested, and maintained just for the purpose of (maybe) making someone feel better. It should have some purpose.
e.g button to leave the app
This seems like a pratical addition, as the logical next step would be to leave the app.
propose him to leave some feedback
If you think there is enough potential value in this to justify the cost of dealing with the feedback you recieve, then sure, add it. Otherwise no.
You say that the exact nature of the selection process can't be explained, but there must be something you describe. Sending a user to a screen that explains the process in as much detail as possible, and specifically informing them of those selection factors over which they have some control, would be a good way to end this experience.