That's not exactly what the page you linked says. The existence of one-way functions implies P != NP, but the reverse is not known to be true. In particular, it's possible that P != NP, but also no one-way-functions exist.

See Impagliazzo's Five Worlds for more: http://blog.computationalcomplexity.org/2004/06/impagliazzos... All five worlds are still possible, given our current results. In both "Heuristica" and "Pessiland", P != NP but one-way-functions do not exist.

I think this is more like Impagliazzo's "Pessiland", where P!=NP but cryptography is still impossible.

I haven't looked at this blog post in detail but the claim seems unlikely. Broadly, P!=NP is about worst case scenarios and the existence of one-way functions is about average case scenarios. If a few paragraphs could prove a relationship between the two, it would be a remarkable result indeed.