I've read numerous accounts about Lincoln peering out over the city from the White House and asking "why won't they attack, why won't they attack?". He was certain the city was doomed. The Union Army was delayed in their reaching Washington and the Confederacy was right across the Potomac in a perfect position to cross over and do as they wished to the capitol, but they never did. They allowed the Northern army to enter and secure the city, thus giving up their best chance of preventing a long drawn out war AND making their secession official. I've read numerous accounts of how they didn't invade, but I've never read an account as to why. Was there ever an official statement made by the leaders of the Rebellion that explained their decision to remain neutral?
Your question presupposes a couple of things that aren't necessarily true. First, the Confederates weren't in any better position to just storm "Washington City" (as it was known then) than the Union was to march straight into Richmond. The early months of the war saw both sides consolidating and, even more importantly, drilling the civilian-soldiers pouring into the respective capitols. Moving an army en mass is an extremely complicated undertaking, and neither side (but especially the Confederates) had the infrastructure in place to get their armies and supply trains organized for a major offensive (let alone a siege). If nothing else, the timeline illustrates this clearly. The war started in April 1861, and the first major clash of the armies came at Manassas/Bull Run in late-July of that year. If it was as easy as just sprinting the 100 or so miles to take the other city, both sides would have tried to do so: but it just wasn't feasible. Either side trying such a thing so early in the war would have been playing with an all-in proposition with extremely green troops (and with no one left behind to guard the capitol just abandoned in case things didn't work out). So in the Spring of 1861, it was a risky proposition at best, and a reckless one at worst.
Second, Lincoln was keenly aware of the strategic and symbolic importance of holding Washington, and some of the first actions he took when assuming office was to reinforce and fortify the city. The accounts you mention of Lincoln saying "why won't they attack" sound vaguely familiar, but I don't immediately recall a source for that. Do you have one? To my mind, that conversation or recollection had to do with the period immediately after 1st Manassas/Bull Run, when the Union was on its heels a bit (though never in any real danger of losing Washington).
The truth of the matter is that both sides in the Spring of 1861 had extremely raw armies with little formal training outside of the officer corps. The armies had neither the infrastructure, numerical strength, or knowhow to pull off so audacious and difficult an operation as sacking the other side's capitol. Even in 1862, when McClellan had a well-trained, sizable army at his disposal, he was unable to take Richmond (a city with far fewer defenders and defenses than Washington). To think that the Confederates could have sacked Washington so early in the war just doesn't jive with the facts and what leaders then (and historians now) understood about the actual capabilities of the Confederate army, such as it was.
[Sources, Bruce Catton, 'Mr. Lincoln's Army'; James McPherson, 'Battle Cry of Freedom'; Doris Kearns Goodwin, 'Team of Rivals']