Let $E$ be a rank two vector bundle on $\mathbb{P}^n$. Assume that $\text{Ext}^1(E, E)=0$. Will $\text{Ext}^2(E, E)$ be zero? Why? Any geometric explanation (in terms of deformation theory?)?


Edit: As pointing out by Angelo, in the case $n=2$, the answer is no. However, I really want to know when $n\geq 4$.



Vanishing of Self-Ext groups of vector bundles
To complement Angelo's answer (which you should accept, as it answers your original question):


If $\mathrm H^1(E^\vee \otimes E) = 0$ then $E$ must be homogeneous, see for instance Theorem 3 [this paper](http://www.math.wustl.edu/~kumar/papers/hydkempf.pdf) by Mohan Kumar.


It is well-known that homogeneous vector bundles on $\mathbb P^n$ of rank less than $n$ splits as a direct sum of line bundles (Theorem 3.2.3 of Okonek-Schneider-Spindler). So if $n>2$ the answer to your question is yes. 


