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As others have pointed out, the question is somewhat imprecise. Here is one precise formulation (of course I do not know if this is what the OP has in mind): "Is there any smooth projective variety $X$ other than $\mathbb{P}^n$ such that for every smooth subvariety $Y$, $\text{det}\ \mathcal{N}_{Y/X}$ is ample?" The OP asks about $\mathbb{P}^1 \times \mathbb{P}^1$ or other Hirzebruch surfaces. Of course these varieties fail the condition. In general, if $f:X\to Z$ is a fiber-type contraction and if $Y\subset X$ is a general fiber, then $\mathcal{N}_{Y/X}$ is $\mathcal{O}_Y^{\oplus r}$ where $r$ equals $\text{dim}(Z)$. So for any $X$ as above, $X$ admits no fiber-type contraction. If one allows $Y$ to be somewhat singular, probably one can rule out all contractions.