In this paper, the $η$-Nash equilibrium ($η$-NE) region of the two-user Gaussian interference channel (IC) with perfect output feedback is approximated to within $1$ bit/s/Hz and $η$ arbitrarily close to $1$ bit/s/Hz. The relevance of the $η$-NE region is that it provides the set of rate-pairs that are achievable and stable in the IC when both transmitter-receiver pairs autonomously tune their own transmit-receive configurations seeking an $η$-optimal individual transmission rate. Therefore, any rate tuple outside the $η$-NE region is not stable as there always exists one link able to increase by at least $η$ bits/s/Hz its own transmission rate by updating its own transmit-receive configuration. The main insights that arise from this work are: $(i)$ The $η$-NE region achieved with feedback is larger than or equal to the $η$-NE region without feedback. More importantly, for each rate pair achievable at an $η$-NE without feedback, there exists at least one rate pair achievable at an $η$-NE with feedback that is weakly Pareto superior. $(ii)$ There always exists an $η$-NE transmit-receive configuration that achieves a rate pair that is at most $1$ bit/s/Hz per user away from the outer bound of the capacity region.