The limiting ratio is $\frac{6}{\pi^2}$.  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 independent and using the Euler product for $\zeta(2)$.