P != NP does not imply anything about the existence of one-way functions. From Goldwasser and Bellare's "Lecture Notes on Cryptography":

However, the above mentioned necessary condition (e.g.: P != NP) is not a sufficient one. P != NP only implies that the encryption scheme is hard to break in the worst case. It does not rule-out the possibility that the encryption scheme is easy to break in almost all cases. In fact, one can easily construct "encryption schemes" for which the breaking problem is NP-complete and yet there exist an efficient breaking algorithm that succeeds on 99% of the cases. Hence, worse-case hardness is a poor measure of security.

Also, two of Impagliazzo's worlds, Heuristica and Pessiland, have P != NP and no one-way functions while two others, Minicrypt and Cryptomania, do.

P != NP does not imply anything about the existence of one-way functions. From Goldwasser and Bellare's "Lecture Notes on Cryptography":

However, the above mentioned necessary condition (e.g.: P != NP) is not a sufficient one. P != NP only implies that the encryption scheme is hard to break in the worst case. It does not rule-out the possibility that the encryption scheme is easy to break in almost all cases. In fact, one can easily construct "encryption schemes" for which the breaking problem is NP-complete and yet there exist an efficient breaking algorithm that succeeds on 99% of the cases. Hence, worse-case hardness is a poor measure of security.

Also, two of Impagliazzo's worlds where P != NP, Heuristica and Pessiland, have no one-way functions while two others, Minicrypt and Cryptomania, do.