If graph isomorphism is in P, is then P = NP?

Question

I think that, since graph isomorphism is not known to be $\textbf{NP}$-complete, we can not reduce all problems in $\textbf{NP}$ to it, and therefore the implication does not hold.

Additionally, in the accepted answer to this question it is stated, that a proof that graph isomorphism is not $\textbf{NP}$-complete would prove $\textbf{P} \neq \textbf{NP}$. Why?