How does Rovelli's idea of time and gravity relate to inertia? In "The Order of Time" (pg 12-13) Rovelli says 

"If things fall, it is due to this slowing down of time. (near a mass) Where time passes uniformly, in interplanetary space, things do not fall.  They float, without falling..."  

So Rovelli is proposing that gravity is caused by the slowing of time; a reasonable hypothesis.
Einstein said "Principle of Equivalence. Inertia and gravity are phenomena identical in nature."  Einstein statement
I have two questions, 
First, is Rovelli's theory on slowing time as the cause of gravity reasonably accepted in the science community?
Second, if yes to the above question, then how does time relate to inertia. It seems to me that there must be a strong relationship between the two.
 A: Yes, for weak gravitational fields the gravitational field is caused by position-dependent time dilation. This isn't Rovelli's theory, it's perfectly standard general relativity found in many books. It's just how general relativity gives back Newtonian gravity.
To see this mathematically, note that for a slowly moving particle the geodesic equation is
$$\frac{d^2 x^i}{d \tau^2} = - \Gamma^i_{00}$$
Assuming the metric is not time-dependent,
$$\Gamma^i_{00} = - \frac12 g^{i\lambda} \partial_\lambda g_{00}.$$
Assuming the metric is almost flat, $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ where $h_{\mu\nu} \ll 1$, we get
$$\frac{d^2 x^i}{dt^2} = \frac12 \partial_i h_{00}.$$
This is just Newtonian gravity in the gravitational potential $-h_{00}/2$. Since the metric component $g_{00}$ determines how fast proper time elapses, we've established Rovelli's statement. 
Intuitively, given initial and final positions and times, a freely falling particle will go between them in the path that minimizes the proper time, i.e. the amount of aging the particle itself experiences. Aging occurs faster in regions with greater $h_{00}$. On the other hand, aging occurs slower when you're moving, so the particle can't just instantly jump to regions with higher $h_{00}$. When you do the math, it turns out the particle compromises by accelerating towards such regions, giving Newtonian gravity.

Inertia and gravity are phenomena identical in nature. [...] Second, if yes to the above question, then how does time relate to inertia. 

Not much. The point is that the gravitational force is 
$$\mathbf{F} = m \mathbf{g}$$
where $m$ is the same $m$ that occurs in $\mathbf{F} = m \mathbf{a}$. Inertia and gravity are related because these two $m$'s are the same, ensuring that all masses get the same acceleration $\mathbf{a} = \mathbf{g}$, which we proved above. Gravity and time are related because the gravitational time dilation determines $\mathbf{g}$, as we also proved above. So both "inertia" and "time" are related to the "$m$" and "$\mathbf{g}$" in $\mathbf{F} = m \mathbf{g}$ respectively, not really to each other.
