In a term paper I wrote for college many years ago, I addressed this exact question. Here's how the POD went:
On that fateful May night in the Wilderness, we have the exact same conditions, that IOTL led to Stonewall's death: a bunch of crazy Tar Heels mistake Stonewall and his staff for Yankee cavalry and fire upon him. However, instead of Stonewall getting hit, his horse bites the dust, shielding him from the hail of bullets. Stonewall survives with only a few bruises and a broken leg. He recovers quickly enough to fight in the Battle of Gettysburg, where a mass Confederate assault on the Round Tops and Culp's Hill shatters the Army of the Potomac on July 2.
After Gettysburg, Robert E. Lee gets sent West, and Stonewall Jackson becomes commander of the Army of Northern Virginia. Lee and E. Kirby Smith recapture the Mississippi River, while Stonewall lays siege to Washington, DC. By the end of 1863, the war is over and the Confederacy is victorious.
However, the fruits of victory soon turn into ashes in the CSA's mouth. On April 14, 1865, Jefferson Davis attends a play at one of the better theaters in Washington (now renamed Jacksonopolis for some reason known only to me
). He is shot in the back in the head by an unstable actor named Booth, and dies the next day.
Davis' successor, Alexander Stephens, is helpless to combat the weaknesses inherent in the Confederacy. Strong support for states' rights, the devastation of the war, political infighting, and the lack of will to stay together lead to balkanization. The Confederacy falls apart in the late 1860s and early 1870s.
Virginia becomes a monarchy under the Lees (Robert's the first king). Stonewall Jackson becomes its first prime minister, serving ably in this position until his untimely death under the wheels of a horse-drawn trolley in 1882.
(If you like what you saw, I can share the rest of the term paper with you, along with a few concepts and ideas I've thought up over the years.)