Does the stability of an LTI system depend on the input?
I don`t hink that we can say that the "output turns out to be unstable". It is the whole system with feedback which is to be considered as stable or unstable (not its output). In the described case (H(s)=1/s), the system reacts upon a step input with a rising output, which corresponds with our expections and with the systems transfer function. Hence, it cannot be considered as unstable. 

(However, we should realize that we speak about an ideal integrating system which cannot be realized in hardware).