I have read recently an article about Lee's retreat from Gettysburg and his pursuit by Meade. It say many interesting things:
http://www.civilwarhome.com/secondantietam.html
So, what might have happened had the river stayed high a few more days? It would have been a battle, the battle of Williamsport, but, who would have won it? And what would have happened next?
Without bothering to review the details - if the very battered Army of the Potomac can summon up enough nerve to attack in force, the extremely battered Army of Northern Virginia probably collapses, losing most of its men and all of its artillery.
The rest of the war: Meade becomes an unassailable hero, until and unless he commits a really ghastly blunder.
Lee and most of his senior commanders are killed or captured.
The AotP spends a month recovering, then advances on Richmond via the Fredericksburg/Chancellorsville route in August.
The CSA reconstitutes an Army of Virginia from the remnants of the AoNV and whatever troops can be scavenged from other theaters. Joe Johnston or even Beauregard comes off the bench to command the new AoV.
This has an immediate effect on the Chickamauga campaign. OTL, the Army of Tennessee was substantially reinforced, including Longstreet's corps from the AoNV. Obviously, ITTL Longstreet doesn't come, but OTL Bragg got other troops (from the Gulf Coast and some other areas) IIRC; ITTL these troops all go to Virginia.
That means no counterattack by Bragg against the Army of the Cumberland. Rosecrans takes Chickamauga, Burnside takes Knoxville, and Tennessee is securely held by the Union, while Bragg retreats into Georgia.
Since they aren't needed at Chattanooga, Grant, Sherman, and the Army of the Tennessee find something else to do - probably an advance on Mobile from Louisiana; the CSA can't really do anything about it.
Meade's offensive stalls in front of Richmond, but pins the AoV.
Rosecrans maneuvers against Bragg in Georgia. Grant comes up to help. Atlanta falls in November.
Yes, this looks much like late 1864, but a year sooner.
I'm not sure how the war plays out from there. Grant and Rosecrans don't like each other, and ITTL they're about equal in prestige. I don't see their armies being combined under one of them, and I don't quite see what the Union would do with two armies in one theater.
Still, it seems most likely that the war ends in 1864.
Longer-term consequences: Rosecrans, who was a Democrat, becomes a very viable candidate for President. The Democrats may even run him in 1864. Grant does not become Commander-in-Chief, go east, and win the final Virginia campaign; he's a much less important figure and probably never becomes President. Meade's bigger and may become President.
PS: Please forgive my English. English isn't my native language.
Nothing to forgive. It's better than most Americans.