Why didn't Lee attack the U.S. Army in the city of Fredericksburg?

Despite the heavy and demoralizing losses the Union army suffered at Fredericksburg, they still outnumbered the Confederate Army by a considerable margin and as a fighting force they were still for the most part in tact, despite being very demoralized.

Before and after the assaults on the Confederate positions, most of the Union Army occupied Stafford Heights, which is on the opposite side fom Fredericksburg. This position was as unassailable as the Confederate positions on the opposite side if not more so. So for Lee to abandon his entrenched position, throw his army across the Rappahannock, and atack the Federals on these hills marching right into the teeth of the artillery they had there would have been insanity.

But the problem for the Union was, the position they occupied, though unassailable, was of little value to the Confederate Army since it was not between them and Richmond or any other key cities and they had no reason to attack it. It was simply a Union headquarter position that held little strategic value. Not only that, the artillery posted there was to far away to be very effective in the battle, and this rendering of the Union Army's best asset as irelevant was one of the biggest reasons for the lopsided defeat at Fredericksburgh.

And while the Confederacy had no reason to attack the North in this position, Burnside was under a great deal of pressure from Washington to attack and defeat the Rebel Army in a certai amount of time. So its probably accurate to say that they were just s much responsible for this Union defeat as Burnside. They basically put him in a position where it was atack now or resign. We all know what happened, so he probably in retrospect would have been better of resigning.

To sum it up, the advantages the Confederacy held in this battle were provided by their entrenched position, and to leave it would have thrown all that out the window. It was a defensive battle that there was simply no viable way to turn into an offensive campaign.