Can this be considered a war crime?
If in this case or in other similar cases the president of the U.S.A. commits war crimes, is it fair to expect that he should be prosecuted? Or are prosecutions only for people in weak countries?
Maybe, but by no means necessarily. The laws of war are designed to be rules that military forces can actually follow; war is a very messy business, and militaries in wartime can't reasonably be held to the standard you'd expect of, say, a civilian police force. So, the laws of war do not include a blanket rule of "don't kill civilians." War will result in civilians dying. It's basically unavoidable. What's required is that countries reduce unnecessary civilian deaths as much as feasible.
There are four ways civilians can die in war as a result of enemy action. Way 1 is that they are known to be civilians and are intentionally killed (i.e. the point of the attack was to kill known civilians). This is obviously a war crime under all circumstances.
Way 2 is that they're civilians, but the enemy thought they were military targets. Additional Protocol I to the Geneva Conventions is the relevant treaty here, and it codifies customary international law about this in Article 57. There is a requirement that military forces do everything feasible to verify at the objectives are legitimate targets. This is not a bright-line standard. It is entirely possible that military commanders would have a legitimate disagreement about whether gathering more information is feasible. And, of course, just because they do everything feasible doesn't mean they'll be right. Mistaken target identifications are not enough to conclude that a war crime happened; that requires looking into information that commanders had at the time or ordering the attack, what other intelligence methods were available and not used, how reliable all the intelligence methods were, etc.
Way 3 is that they were direct collateral damage in a strike. Article 57 again provides guidance. There is no zero-collateral-damage rule. The rule is that militaries must take all feasible precautions to minimize and if possible avoid collateral damage, and that militaries may not attack unless the anticipated military advantage outweighs the anticipated collateral damage. Again, it depends on the information available to commanders at the time, and the options they had. It is not necessary to send in snipers to eliminate collateral damage if that would be infeasible.
Way 4 is that attacks result in indirect collateral damage. For instance, if key parts of the water infrastructure in a city are destroyed, that can result in civilian deaths long after the attack. These have special protection, and you can't intentionally target food or water infrastructure (even if it is helping the enemy military as well) if destroying it would leave the civilian population with inadequate food or water. But way 4 isn't really what's happening here.
You may notice a theme. The Geneva Conventions grew out of customary international law, which is a fancy way of saying "what countries actually did because they felt they were obligated to." They were written with the hope that militaries would follow them. That means they couldn't take absurdly extreme stances like "the death of a civilian is automatically a war crime." The goal is to minimize harm to civilians; it's simply not possible to eliminate it.
It is entirely possible the US drone program involved war crimes. However, it is impossible to make the evaluation without knowing (highly classified) details about US intelligence and military capabilities, and the knowledge commanders had when they made their decisions.