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, but it might be easier to read this 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 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) 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) 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 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. More recent related papers include this one by Andreatta-Wiśniewski, this one by Araujo and this one by Araujo-Druel-Kovács.