In actual fact though, Meade wanted to withdraw, before Lee had a chance to attack him(Meade, certainly, had no desire or plans to attack Lee)and continue McClellan's campaign on the Peninsula.
It seems, to me, that Meade's plan in Va. was to keep about 24 hrs. distance between himself and Lee. Lincoln can be forgiven for thinking that the offensive war in the East, that he wanted, was not really Meade's cup of tea.
Actually Meade wanted to stop the war, for a little R&R for the AoP and then dust off McClellan old Richmond campaign., both an anathema to Lincoln and Stanton. Meade, like many AoP commanders, never seemed to realize that the wishes of Lincoln was, and would be, a significant factor in any of his plans.
In the time between Gettysburg and Grant's arrival in the East, did Meade bring Lee to battle?