In short, no, but there were certainly spectators during the earliest engagements, especially at the First Manassas/Bull Run, and during a few prominent spectacles like the showdown between the Ironclads. The popular perception among the intellectual and political elite in Washington was that the war would be over quickly, likely after a few pitched battles during which the rebels would break until Richmond was captured, ending the conflict.
At Bull Run, there were many spectators, predominately male, that traveled over 25 miles from Washington, DC, a 7 hour trip, to observe. Described as a "throng of sightseers" by Captain John Tidball, a union artillery commander, the crowd, composed of people from all walks of life, "from the most grave and noble senators to hotel waiters" set up on hills some 5 miles behind the lines, the Centreville Heights, where at least one woman attempted to view the battle through opera glasses. Ultimately, they just saw a lot of gun smoke from their vantage points.
Dissatisfied both with hour-old news about the battle and the poor view, a handful of politicians, including some senators, and prominent citizens, including famed photographer Matthew Brady, to push closer, all the way to a field hospital at the rear of a division's line. When that line eventually broke, those civilians were caught up in the panicked retreat. One senator, Elihu Washbourne, attempted to rally the faltering troops while another Alfred Ely, was so close that he got himself captured by confederate cavalry and spent 5 months detained in Richmond.
So yes, there were many people spectating, mostly men and few women, either high class "ladies" or "hucksters" selling food, but calling them picnickers as we might imagine today would simply be inaccurate; folks had to carry provisions with them for the long journey into what was now enemy territory, a trip which was in not a pleasant outing but rather an adventurous excursion. Only a few of those men were actually caught up in the hasty retreat, though all made their way back to Washington, and none were actually under fire with the exception of the governor of Rhode Island, who had two horses shot out from under him.