On the question of whether McDowell's army could have mounted a post-victory pursuit: some people

are determined to ignore whatever anyone else says and insist "it's impossible, it's impossible, it's impossible," even though these same people think Johnston's and Beauregard's Confederates, who would serve as our model for what a green army could do in the wake of a win at Bull Run, could have pursued McDowell back to Washington.
I'll state this one more time, just to make it crystal clear:
neither McDowell nor any of his senior subordinates would have even thought of ordering a pursuit. Like their opposite numbers in the Confederate army, at Bull Run there was no one of senior rank with the ruthless aggression to push aside the difficulties and plow forward, no one who would think that keeping the initiative and giving the aggressive a try was better than doing nothing. You can' construct a scenario where enough people get killed or wounded to put such a person in charge without beheading the army -- either army, really.
But as for the larger question of whether it was possible? The overwhelming majority of historians who have voiced an opinion on the matter think that the Confederates could have mounted a pursuit to the Potomac, which was some 30 miles away. Given that the differences in quality between the equally green armies were minute, a point proven by how close run that battle was and countless studies since then, the idea that the Federals couldn't attempt at least as much is an exercise in willfully capricious judgement. And that makes it automatically invalid.
Some people

are also very eager to brush aside the issue of command, and to that I will ask this one simple question: what do you think might have happened if you could wave your magic wand and put a Grant, Sherman, Lee, Jackson, or Sheridan in command... of
either army. Or even a Rosecrans or Early? Do you think they would have sat their twiddling their thumbs the next day? Hell no -- regardless of circumstances, they would have pursued. Some of those guys would have started their pursuit in the wee hours of the morning, before sunrise.
Maybe it would have worked, and maybe it wouldn't, but this is what separates the better commanders from the lesser ones: when faced with a glittering opportunity of an enemy on the run, they don't stop to regroup and pull up their logistical tale -- instead, they
get going and go after them!
And this is why I say if the posters who fret about it and say it was impossible would have made decent aides on McClellan's staff. They would have fit right in!

In this way, you can literally see how the command tent of a McClellan or Johnston must have operated, because here we have people who are under no pressure whatsoever, and they are so blinded by the problems that they can't see the opportunities and actively argue against reaching for gold.