It was always the Union's war to loose and despite some rough patchs the Union never permanently lost territory. No war is won on the defense but I welcome any major examples in the last 150 years to prove me wrong.
It's generally not considered valid to argue from "this happened, therefore nothing else could happen". If you have a look at my post, you'll see I outlined several options. One of them is a two-step process.
Step one: the Union loses the Army of the Potomac totally in the Seven Days.
Step two: A Confederate offensive into Maryland in July.
Once a large CS army is in Maryland, the Union has a choice of going out to face it or defending Washington - they don't have enough troops left in the East (once the AotP is gone) to defend Washington and still send out a large enough force to stop Lee's army. Lee can then, depending on what the Union does (defend Washington, fight him, defend Baltimore and Washington...) react to their movements. Washington is certainly not invulnerable in this period, there's several avenues of attack and defending them all with the right amount of force is a matter of luck.
Historically it took McClellan and a large chunk of the transferred Army of the Potomac to stop Lee. With both McClellan and that large chunk of the Army of the Potomac lost then the Union is in serious trouble - if the Union had an army large enough to beat Lee without including any of McClellan's Army of the Potomac, it begs the question about things like Bull Run (or for that matter why McClellan didn't get his darn reinforcements).
This is, of course, an offensive.
As for your argument about wars being won or lost on the defensive, I'll bring up the Winter War as a microcosm. That's a war where the USSR was convinced by a vigourous Finnish defence to leave most of Finland alone - the Civil War analogy would be a successful defence in Virginia to the point that the Union went for peace in 1863-4.
The other way for the CSA to become independent is a European intervention. If it's Britain, especially if it's Trent, then it's very hard indeed to see the CSA
not becoming independent!