So the fact that you don't see anything about the speed of gravity in Newton's equation is a bit of a clue; Newtonian gravitational interactions propagate instantaneously (ie. with infinite speed). If there was some finite speed of propagation, we'd have to have a constant related to that speed in the force law, and we don't.
It's a little more obvious to see if you write Newtonian gravity in terms of Poisson's equation instead of Newton's force law;
$\nabla^2 \Phi(\textbf{r},t) = 4 \pi G \rho(\textbf{r},t)$
In case you've not seen Poisson's equation before, it's the field equation for gravity. That means that given a distribution of mass $\rho(\textbf{r},t)$, we can calculate the gravitational potential $\Phi(\textbf{r},t)$. From $\Phi$ we can calculate the force of gravity at each point, and if we place a point mass at the origin we recover Newton's force law.
The reason that it's obvious from Poisson's equation that Newtonian gravity is transmitted instantaneously is because it doesn't have any time derivatives ($\nabla^2$ contains only spatial derivatives). We solve the equation to calculate a spatial distribution of the gravitational potential at a given time, and then if the matter distribution in our spacetime changes, the gravitational potential changes instantaneously at all points in space.
This was the main motivating reason for Einstien's search for a relativistic theory of gravity.
There's a lot more information in the wikipedia article called 'speed of gravity' if you're interested.