Is that possible to calculate modular inverse of a point on elliptic curves? Imagine that you are given a point $P$ so that $P=a\times G$. If you have no knowledge of $a$ is that possible to calculate point $I$ so that $I$ is the modular  inverse of $P$?

We know that over prime fields each member has a modular inverse, which means for $a\in \Bbb F_n$ there exists $x$ so that $ax\equiv 1\pmod n$. Now, My question is, is that possible to calculate point $I$ on the same curve as $P$ for which the equation  $I=x\times G$ holds. This is to happen with no knowledge of either $a$ or $x$. 

If such algorithm exists or become discovered in future will that be considered a weakness to the security of elliptic curves?
No, this is a known variation of the computational Diffie-Hellman problem, called Inverse computational Diffie-Hellman (InvCDH), according to which on input $g$, $g^{x}$, it is difficult to find $g^{x^{-1}}$.

See section 2.2 at Variations of Diffie-Hellman Problem by Bao et al. for more info and problem reduction proofs.