A system having impulse response $ h(t)=u(t) $ stable or not?
Your derivation is correct. The impulse response is not absolutely integrable, hence the system is not BIBO stable. From the corresponding transfer function $H(s)=1/s$, you can see that there is a single pole at the origin. Systems with single poles on the imaginary axis, like the integrator in your example, are also called marginally stable. These systems are not BIBO stable, transients do not decay, and certain inputs can cause unbounded outputs.