Since you are located in Germany, information about US laws is misleading at best.
According to German employee protection laws, you can actually fire an employee for being constantly sick under very strict conditions none of which I see in this case (in short, a negative medical prognosis is vital, i.e. a doctor stating that it's going to get worse, not better).
There is nothing in the law that I recall which outright prohibits speaking about this topic, but you are definitely entering a minefield.
At the same time, there is a small chance that the medical problems might be related to his work, maybe caused by stress or an unhealthy work environment or something you are not even aware of (e.g. mobbing). In such a case not asking could be problematic as well as you have a duty of care for your workers.
The first thing you should do is discuss this question with HR and possibly involve the works council (Betriebsrat) as well, if there is one. IMHO your best approach would be to tell him that you noticed he is sick at times, reassure him that HR has no complaint about it and that you bring the topic up because you are worried if it might be related to the job and if he wants to talk about it so you have a chance to make any changes that might improve the work environment for him. Explicitly offer him that if he doesn't want to talk about it, that is fine as well, but you want him to know that if it is work related in any way, you will be happy to hear about it - now or later - so that you can look into it.
You can also pick up the advice from Chronic Pain Sufferer and tell him that even if he doesn't want to talk about the sick days, if he is suffering from something chronic, you don't need to know any details, but he can tell you whatever accommodations you could do that would make his life easier.
Essentially, the idea is to approach this as an opportunity to support a person who might be having a difficult time and really doesn't need more trouble - but would be thankful for any support.
Do not under any circumstances approach this topic without the support of HR, and I strongly propose involving the workers council as well.