PRG existance and P versus NP
First, note the following:


If one-way functions exist then P!=NP (src)  
Pseudorandom generators exist if and only if one-way functions exist (src)


From there the proof is trivial.

Now, I wanted to explain (or at least try as this stuff is not my expertise) why the reduction you have in the question is not sufficient.

NP does not mean not-polynomial time, it means nondeterministic polynomial time. Your reduction argues that there is no poly time algorithm. That says nothing about whether or not there is a nondeterministic poly time algorithm. You would have to first show that there is a nondeterministic poly time algorithm, then show that there is not a poly time algorithm (or vice-versa). There may be other problems with the reduction in the question (again this is not my area of expertise). For example, the argument seems to be circular. Hopefully others can comment on this.