At most Lee would have gotten to the outskirts of Washington, to the forts manned by the Heavy Artillery units. The idea that the Army of Northern Virginia could have taken Washington, DC in 1862 is pure fantasy. Just as Richmond took a long seige to take, so would have Washington, and the Confederate Army could never have won on a protracted seige because the Union reserves would be mobilzed to break it before it could succeed. Lee's retreat sfter his invasion would always provide the situation for Lincoln to announce the Emancipation Proclaimation and once that Proclaimation was received in London, any chance that London would recognize the CSA before Washington did was gone.
Lee’s Biographer Douglas Southall Freeman said that Lee’s greatest chance to destroy completely a Union Army came in the Battle of the Seven Days. At Glendale Lee had already mauled the Union Army of the Potomac and sent it reeling back from the gates of Richmond, “Stonewall” Jackson’s “Army of the Valley” had finally arrived and now moved to facing the weak Federal right flank which was accorded along an area of swamp land, the next day Lee ordered classic double envelopment with James Longstreet’s Corps attacking from the South against the Federal Left and Jackson from the North against the Federal Right, the Federal Commander George B McClellan had abandoned the field and fled back to his headquarters on the banks of the James during the night and the Army of the Potomac lacked any leadership with Divisional commanders largely left to fend for themselves, Longstreet’s attack went well but for some reason Jackson’s attack never materialised and Longstreet was forced to end his own attacks, the reason for Jackson’s tardiness at Glendale can largely be attributed to exhaustion he had been deprived of any real rest for weeks and at the same time had lead perhaps the finest campaign of the civil war (“the Valley Campaign” of 1862), so suppose that Jackson gets some rest and attacks?
Jackson’s Corps smashes through the Federal Right flank and soon after Longstreet does likewise on the Federal Left, with both Federal flanks torn away it is left to every divisional commander to get out as best they can, the only two who might have achieved this would have been Joseph Hooker who was commanding a division slightly to the South and Phil Kearny who was perhaps the finest commander in the Army of the Potomac at the time (he was later killed at Canntily during the Federal retreat from Second Manassas.
Such a victory would have rendered 2/3rds of the Federal Army destroyed and the Federal Base at Harrison’s landing vulnerable, Lee would probably have easily seized the Federal base along with McClellan’s vast artillery train (which had been intended to reduce the fortifications of Richmond), Kearny and Hooker may have been able to withdraw their troops leaving either French or Sumner to either surrender Harrison’s landing or die defending it, but still that would leave but 20,000 men with little heavy equipment of over 120,000 men who had landed there at the beginning of the campaign. Lee would have moved North. John Pope may still have taken his patchwork command of the “Army of Virginia” out to meet Lee and would have been soundly beaten in fact with a probably noticeably smaller command Pope may well have been completely annihilated by Lee in the area around Manassas leaving only very limited Federal forces within the City of Washington with freshly raised troops forming the bulk, in such a situation Lee could have invaded the North as Braxton Bragg and Kirby Smith invaded Kentucky in Maryland he would have found little to stop him, with either Hooker or Kearny in command of the forces in Washington they would have realised that a force of what would have been about 40,000 could not hope to smash Lee and would have urged that Union forces from secondary theatre such as South Carolina, North Carolina and perhaps even Louisiana be transported to Washington to reform a credible Army this would have left ports such as Charleston, Williamsport, Savannah, Mobile and even perhaps New Orleans under far less pressure and possibly permitting far more foreign aid to the South to enter the CSA, While at the same time confederate forces from these areas could be moved to Kentucky or Maryland, with Lee free to exploit the rich farmland of Pennsylvania and Maryland and with the likelihood that on entering the Pro-Southern East of the Maryland his Army would have been reinforced with new recruits while at the same time receiving men from along the Eastern Sea Board who had been freed thanks to the Union withdraw from these areas. In Kentucky with Lee dominating the Eastern theatre and reinforcements coming in all the time Bragg and Smith would probably have been emboldened to continue their offensive after the indecisive battle of Perryville and would probably have been able to compel General Buell to withdraw back into the fortifications of Louisville while Bragg’s plans for installing a confederate government in Kentucky to raise troops could have gone a head and within a mouth or so may have furnished the Confederate Army of Tennessee with an additional 5,000 or so men as could well have been the case with Lee in Maryland.
With the Confederates dominating both East and West and only Rosecrans and Grant holding their own at Corinth Mississippi against the Confederate General Van Dorn it is likely the English government (PM Palmerston had said after second Manassas that a second Confederate victory would compel Britain to recognise the South and France was already at this stage very eager to do so), Lincoln faced with the Royal Navy breaking up the Union blockade and with the threat of a British army in Canada (which including Militia number 110,000 men, spread wide but still a powerful force if concentrated) would have been forced to “acquiesce to the good entreaties of the European power” and may well have resigned leaving Hannibal Hamlin to suffer the repercussions of a treaty with the south which may well have left Eastern Maryland as part of the CSA (the western part of the state remaining in Federal hands and providing a corridor of land around washing DC itself) and Kentucky (or at least the greater part of it) under confederate control). Within a decade or so the Economic ties that would have grown between North and South would have trumped any lingering hostility and while their may have been a friendly rivalry and some suspicion (with the CSA probably unabashedly allied with France and England) relations would not have been very bad between the two nations. However Slavery (Which would probably have come ot a slow end between the late 1880’s and mid 1910’s) would have been a bone of contention.
