This is an interesting question. I don't know if anyone has thought about it, but it seems natural from the point of other results. I don't have a proof either way, but I have some ideas:

The following generalization of the Kobayashi-Ochiai theorem of Cho--Miyaoka--Shepherd-Barron (3 people) may be relevant:
> **Theorem** Let $X$ be a (smooth projective) Fano manifold of dimension $n$ and assume that $c_1(X)\cdot C\geq (n+1)$ for any (proper) curve $C\subset X$. Then $X\simeq \mathbb P^n$.

You can find the original article in [this book][1], but it might be easier to read [this][2] proof by Kebekus. 

Another possibly relevant statement is the following
 > **Proposition** Let $X$ be a smooth projective variety of dimension $n$ and $\mathscr L$ an ample line bundle on $X$. Assume that for some $d\in\mathbb N$ there is a point $x\in X$ such that any general point $y\in X$ can be connected to $x$ by an irreducible rational curve $C_{y}$ with $c_1(\mathscr L)\cdot C_y \leq d$. Then $c_1(\mathscr L)^n\leq d^n$.

This is much easier than the above. It is proved for example in V.2.9 in [Rational Curves on Algebraic Varieties][3] by J. Kollár. 

As promised, this does not give you a proof, but suggests that there might be some interesting statement. Namely, by *Bend & Break* (see II.5 of [Kollár's book][4]) for every $x\in X$ there exists an irreducible rational curve $C\subset X$ such that $x\in C$ and $c_1(X)\cdot C \leq (n+1)$. In other words, the Cho--Miyaoka--Shepherd-Barron theorem says that if $(n+1)$ is indeed the smallest value possible to satisfy this inequality, then $X\simeq \mathbb P^n$. 

On the other hand, if there is a point for which the minimal degree rational curves going through that point cover a dense part of $X$ and this minimal degree is $d< (n+1)$, then by the above Proposition it follows that $c_1(X)^n\leq d^n < (n+1)^n$, so $X\not\simeq \mathbb P^n$. Of course, it is possible that the minimal degree rational curves do not dominate $X$, but Mori's proof of the Hartshorne conjecture (see V.3.2 of [Kollár's book][5]) suggests that it might not happen for such high minimal degree. 

Anyway, this is getting pretty long for an answer without an answer and so far I have only addressed the projective space case and not the quadric. For that the situation is similar: there is a [paper][6] by Miyaoka proving the equivalent of the above Theorem for quadrics and the pseudo-argument about minimal degree rational curves applies the same way. 

There are also plenty of related results. Some are referenced in the Cho--Miyaoka--Shepherd-Barron paper as well as in Kebekus's. The background to all of this is included in [Kollár's book][7]. More recent related papers include
[this one][8] by Andreatta-Wiśniewski,  [this one][9] by Araujo and [this one][10] by Araujo-Druel-Kovács.




  [1]: http://www.ams.org/bookstore?fn=20&arg1=aspmseries&ikey=ASPM-35
  [2]: http://link.springer.com/chapter/10.1007/978-3-642-56202-0_10
  [3]: http://books.google.com/books?id=oqW3GabJLjgC&lpg=PP1&pg=PP1#v=onepage&q&f=false
  [4]: http://books.google.com/books?id=oqW3GabJLjgC&lpg=PP1&pg=PP1#v=onepage&q&f=false
  [5]: http://books.google.com/books?id=oqW3GabJLjgC&lpg=PP1&pg=PP1#v=onepage&q&f=false
  [6]: http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=198827&vfpref=html&r=10&mx-pid=1929792
  [7]: http://books.google.com/books?id=oqW3GabJLjgC&lpg=PP1&pg=PP1#v=onepage&q&f=false
  [8]: http://link.springer.com/article/10.1007/PL00005808
  [9]: http://link.springer.com/article/10.1007/s00208-006-0775-2
  [10]: http://link.springer.com/article/10.1007/s00222-008-0130-1