Almost certanly not. Had Lee won at Gettysburg he would still have been in the middle of Union territory. He would have used up much of his ammunition with no line of supply back South. He would have had tens of thousands of wounded to care for. And D.C. was ringed with forts and had a garrision as large as, if not larger than his remaining army. All Lee could do, win or lose, was go home.
LeoMcNeil,
I think that DoodleDawg’s post also leads to another facet in all this. The military strategists of the 19th century invariably focused upon enemy cities as targets. This was a mistaken notion. The real target should have been the enemy army, not the enemy’s cities. I think Lincoln recognized this fact fairly early on in the war. If an attack on DC were imminent, the government would simply have evacuated the capital and moved elsewhere.
Political opposition would have grown, but Lincoln was not up for reelection until Nov 1864, a full 16 months after Gettysburg. Assuming a loss at Gettysburg, the Army of the Potomac had prepared defensive lines to which they were ready to retreat. These lines were intentionally placed to guard DC (and Baltimore). How long would it have taken Lee to break through those lines and continue to threaten DC, especially given his supply situation? Lee was never truly a threat to DC, even with a Confederate win at Gettysburg.