Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$
I summarize the first two pages of Kedlaya, A construction of polynomials with squarefree discriminants
When $P$ is irreducible and the discriminant $\Delta(P)$ is square free, the number field $\mathbb{Q}[x]/P(x)$ has ring of integers $\mathbb{Z}[x]/P(x)$... When the coefficients of $P$ are chosen randomly, this is expected to occur with probability $\prod_p a_p$ ... where $a_p$ denotes the probability that $\Delta(P)$ is not divisible by $p^2$. These probabilities have been computed by Brakenhoff:
I'll omit the table, but they are all of the form $1-O(1/p^2)$, so the product is nonzero.
Unfortunately, ... it seems quite difficult to prove that a polynomial takes squarefree values with the expected probability ...
Keldaya then summarizes work of Granville and Poonen which, assuming the ABC conjecture, implies that $\Delta(P)$ is squarefree with probability $\prod a_p$, where the limit is taken over boxes in $\mathbb{Z}[x]_{\deg n}$ of a certain shape (roughly, much longer in one direction than the others.).
However, without assuming any conjectures, it is difficult to establish even the existence of infinitely many polynomials of a given degree with squarefree discriminant.
Kedlaya then explains that constructing infinitely many such polynomials is his main result.
In short, there is a good conjecture for the probability of squarefree discriminant, but people can't unconditionally show that it is even positive. Squarefree discriminant is a bit more special than $\mathbb{Z}[x]/P(x)$ integrally closed, but I think this is suggestive.
The conjecture mentioned in the comments was proven in 2016 by Bhargava--Shankar--Wang:
The density of monic irreducible polynomials $f$ with integer coefficients such that $\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction field is $\zeta(2)^{-1} = .607927\ldots\,\,\,$.
This is Theorem 1.2 of Squarefree values of polynomial discriminants I, https://arxiv.org/abs/1611.09806. An updated version is at http://www.math.uwaterloo.ca/~x46wang/Papers/monicsieve4.pdf.
Note that the density they use is with respect to the height, $$H(x^n+a_1x^{n-1}+\cdots+a_n):=\max(|a_1|,|a_2|^{1/2},\ldots,|a_n|^{1/n}),$$ which is a little different from the one proposed in the question on this page.