Meade attacks, but later!
First off, you'd have to change all of the dispostions of both armies. Lee was able to have two of his corps on the field with a great advantage in their deployment - that being Ewell's corps came down behind the Union line facing Hill's corps (albeit not essentially planned that way). The next is more of the Union army would have to arrive on the first day too, not only hold off Hill, but far more importantly you'd have to hold off Ewell's attack. But the Union didn't have such a situation, so the Union couldn't attack due to the South's deployment not to mention that the South enjoyed the numbers in all the right places.
From the results of the fighting on 1st July, Lee then takes the initiative because of his victory. The Union army really doesn't come together as such until late on 2nd July, by which time they're well & truly on the defensive. Ironically, thanks to Lee's efforts, Meade is fighting the battle the way he wanted. Lee, thinking victory is just a matter of one more attack, throws all caution to the wind, ignores the concerns of Longstreet, & thus witnesses Pickett's chrage cut down in a manner similar to what happened to the Union army at Fredricksburg. Up until this point, though, Meade isn't in a position to attack. But now he is. So now is the chance for your scenario to come about.
The Southern Army is bruised, battered & clearly defeated after three days of battle. The Union army, although bruised & battered, is in better shape. Meade's time is now, but he doesn't take the chance, even though the likes of Hancock, Doubleday, Howard & Pleasonton all wanted to attack. So if Meade took their advice, would have Lee been thoroughly defeated in the counterattack along with an end to the civil war?
Other than that, yes there were other Union forces around, most notably those covering the Washington to Baltimore area, which were of a respectable number. In otherwords, the Union casualities at Gettysburg, should have Meade lost & been forced to withdraw, would have been replaced within a week. Lee, however, never had such reinforcements. So a second battle outside of Washington could have seen a much smaller Southern army, of say 50 000 troops, vastly outnumbered by 120 000 Union troops, about a third of which are fresh.