That's where you are wrong. In short - you can't look at a single product of the reaction, you have to look at everything that happens.
Imagine you have a reaction with more than one product, it may happen that one of these products is highly stable (like water, or carbon dioxide, or sodium chloride) and its stability is what is driving the reaction - but other product is much less stable.
It may also happen that the reaction is driven not by the stability of the products, but by the increasing entropy of the system.
Note that I used term "stability" rather lousily, just to answer the question in terms you have used.