The limiting ratio is $\frac{1}{\zeta(2)} = \frac{6}{\pi^2}$.  A proof can be found for instance in $\S$ 24.10 (of the sixth edition, but I don't think that matters) of Hardy and Wright's *An Introduction to the Theory of Numbers*.  See $\S$ 7.3 [of these notes][1] for the statement of a generalization to the primitive lattice point enumerator for regions in $n$-dimensional Euclidean space.  (The proof was left to two students in the graduate seminar I was then running.  They presented a proof, but I didn't get around to incorporating their arguments into the writeup.  Anyway the transition from the two-dimensional case is straightforward.)  


This is a standard result in the elementary Geometry of Numbers, equivalent to the problem in elementary number theory asking for the probability that two positive integers are relatively prime.  You can see intuitively that the above answer is correct by considering the events "$a$ and $b$ are both divisible by $p$" -- as $p$ runs over all primes -- as independent and using the Euler product for $\zeta(2)$.

<b>Added</b>: In $n$-dimensional space the same reasoning leads to the answer is $\frac{1}{\zeta(n)}$.  (Which of course means that it is in some sense unknown exactly what the answer is whenever $n$ is odd...)


[1]: http://math.uga.edu/~pete/geometryofnumbers.pdf