The Civil War was still going on at the time of Lincoln's assassination (April 14, 1865). Lee's army in Virginia had surrendered, but the various other Confederate forces had not. So the Union couldn't really take up arms again - they hadn't yet put down their arms!

In the weeks and months after Lee's surrender on April 9, 1865, Confederate armies surrendered in piecemeal fashion. The most notable of these were Johnston's surrender to Sherman (Carolinas) on April 26, Taylor's surrender (Alabama) on May 4, Kirby Smith's (Trans-Mississippi Front) on May 26, and Stand Watie's (Native Americans) on June 23.

The "end of the war" is generally dated as May 9, 1865. On that day, President Andrew Johnson issued a proclamation declaring the rebellion "virtually at an end".

However, not all forces had surrendered by that point, and various minor rebellions continued to pop up in the following months. It was not until August 20, 1866, more than a year after Lee's surrender and Lincoln's death, that President Johnson proclaimed the official end of the war.

Now, therefore, I, Andrew Johnson, President of the United States, do hereby proclaim and declare that the insurrection which heretofore existed in the State of Texas is at an end and is to be henceforth so regarded in that State as in the other States before named in which the said insurrection was proclaimed to be at an end by the aforesaid proclamation of the 2d day of April, 1866.

And I do further proclaim that the said insurrection is at an end and that peace, order, tranquillity, and civil authority now exist in and throughout the whole of the United States of America.

In testimony whereof I have hereunto set my hand and caused the seal of the United States to be affixed.

Done at the city of Washington, this 20th day of August, A.D. 1866, and of the Independence of the United States of America the ninety-first.

Put simply: there was almost no one left in the south to fight one. By April, 1865 the south had all-but exhausted its manpower, navy, transportation network, munition and food stores, which was all aside from the fact that the Confederate government was on the run. The armies the Confederates did have in the field were dangerously undersupplied, outmanned by overwhelming opposition, and without a central government apparatus to keep the war going on any practical level. For Lee, when he surrendered (just before Lincoln's assassination), he was outnumbered nearly 4:1 by Grant, Meade, Sheridan, and the Army of the Potomac. Joseph E. Johnston wasn't looking at much better odds against Sherman when they met at Bennitt Farm just AFTER the assassination, and despite Jefferson Davis' order to Johnston to escape with the cavalry and continue fighting piecemeal, Johnston decided to surrender.

Johnston was on the ground and had a no-B.S. understanding of the military reality with Lee out of the picture. As a practical, experienced soldier with a lifetime of experience under his belt, he knew that any continued fighting was pointless. The Confederate armies had no infrastructure to support them, a collapsed navy, and there weren't any more recruits coming: it was over. Even worse, with Lincoln's murder, the Union was invigorated with a vengeful attitude, so it wasn't like they were going to sue for peace any time soon.

By mid-April 1865, the war was over for all intents and purposes (official victory would happen a few weeks later), and short of foreign intervention, there was nothing the Confederacy could do to keep things going.

[Sources: Jay Winik, 'April, 1865'; Bruce Catton, 'A Stillness at Appomattox']