So, part two of my response.
Centerville is 3 to 5 miles from the battlefield dependent on route (most likely 3 to 4 miles). Fairfax Station and the 4th Michigan are at most another ten miles further on (I don't know their precise location – Fairfax Court House and Fairfax Station – both of which they are said to be at – are not quite the same place). It's another fifteen or so miles to the Washington Defences. So, that's c.30 miles from the battlefield to the outskirts of Washington... assuming that the Southern army takes the most direct route.
Now, let's look at the state of the Confederate Army. Evans, Bee and Bartow's brigades have been shattered. Jackson's brigade has been heavily engaged while Cocke, Elzey and Early's brigades as well as the Hampton Legion have also been engaged. That leaves as entirely fresh Bonham, Ewell, D.R. Jones, Longstreet and Holmes of the Army of the Potomac. That's c.20 regiments of totally fresh infantry and a minimum of 20 guns (There is some question over precisely how many guns the Washington Artillery had at this point).
So, let's assume that Bonham and Longstreet showed more desire for the attack than they actually did and that Bureaugard and his staff were able to send less confusing orders than they previously had been doing then there is a good opportunity to get 20 regiments to face 8-12 regiments at Centerville. Union troops that has just had the rest of its army rout through them and who are likely to be understandably nervous. Or at least that is the theory.
Bureaugard/ Johnston need to police the battlefield. Round up prisoners... and there are lots of prisoners, gather discarded guns and rifles and so on. Now logically the units that had been most engaged should do this but staff work on both sides up to this point had been poor so there is no certainty that it would have been divided up like that. Furthermore the (barely adequate) roads are clogged with panicked civilians. Civilians that badly hampered the retreat of the Union army and are likely to have a similar impact on the Confederacy. They are also likely to create even more prisoners that need to be guarded. Therefore Centerville may only be a few miles away but is the possibility of capturing it illusory?
Well, the battle ends around 5PM. Let's assume that Southern troops are prompt to follow. Even so a battle for Centerville can't start until 6PM at the
very earliest (McDowell had The US 'Regulars' hold in place and try and protect the retreat and getting past them would have taken time). Again let's assume Centerville is taken. What now? Washington. That's still maybe twenty five miles away. The Battle of Centerville must have taken time. Lets be arbitrary and assume the Union basically collapses. It will still cause a delay... and more prisoners. So, twenty minutes/ half an hour is lost there. So, it's 6:30ish, the roads are clogged and they have at most four hours of light left (looking at
www.timeanddate.com) That requires six miles an hour. Doable? For cavalry certainly but these are volunteers. Fresh troops.
So is a more accurate question, could what (little) cavalry the South had have taken Washington and its 18 fresh infantry regiments assuming that the infantry are able to coordinate sufficiently to defeat the forces at Centerville. In a word... No. No chance. The South are cavalry light (not as light as the North, but still cavalry light) and have maybe 1,500 horseman and no 1,500 horsemen ever formed could shift 18 fresh infantry regiments from behind fortifications.
I am sure they could have caused panic and more efficiently attempted to pursue the 'retreating' army but I think it is perfectly understandable why they don't. They are split up in to many little packets and only Stuart has a formed command of any size – and they had already attacked and been repulsed by Union infantry (in so doing slowing down the Union troops and allowing the Confederates time to create a new defensive line).
All that said then, what about an attack on Washington the following morning? That seems more reasonable. However now one has given time for the rest of the Union Army of 'Northeastern Virginia' to attempt to reform...
Therefore despite people like Jackson thinking they should have pressed on I think that the South would have found that Washington was a hard road to travel...
[Obviously I am more than interested in hearing alternative views]