I guess most here would agree that Grant wouldn't have been content to let Lee slip away, as McClellan was in Maryland. But even Meade exercised a fairly agressive pursuit once he realized that the ANV was heading south.
So the first question is, if Grant had been in charge on July 4th, would he have counter-attacked, and if so, how and where? It's quite possible that the positions of the army were such that no offensive action was likely to yield a positive outcome. The flanks had been re-enforced from actions the first two days, and in the center there was simply too much open ground for the infantry to charge - in either direction - without suffering too many losses from cannon fire directly ahead and from fire on either of the flanks.
So what happens if, after Pickett's charge fails, Grant makes a counter-charge, but it takes too long to organize it (a half-hour or better)? Would he be stopped by re-supplied Confederate guns and a re-organized Confederate line on the ridge adjacent to Seminary Hill? If so, what would be the result? I'm guessing that ultimately the result would be the same - the Confederate forces would have to retire, they couldn't stay consolidated in one place for very long due to supply and forage problems. But the additional Union losses would have made it seem much more of a "draw" in public opinion, rather than a defeat for the Confederacy.
And if Grant decides not to counter-attack across much the same ground, but to attack once the ANV begins a withdrawal, how is that much different from what Meade did historically? Would not the results be the same? What further or additional steps could Grant have taken at that point to overtake and stop the Confederate army's retreat into the safety of Virginia?
Bringing the