RH would imply g(p)=O((log p)*p^(1/2)), not p^(1/2).
I don't know whose conjecture your g(p)< ln(p)^2 is refering to? There's Cramer's that says the lim sup of g(p)/log(p)^2 is 1 but this doesn't imply your inequality.
What's a "determinate prime"? Knowing the largest gap between p and p^2 is p implies the largest gap less than p is p^(1/2) (just consider a prime larger than sqrt(p) and apply this again, and repeat). Since this is quite a large leap from current results (and stronger than what RH implies) you're going to have to provide something stronger than "My calculation show's it's true" before I come near believing you can prove this.