Potomac River*
One of the first actions by the Union were to push troops across the Potomac and seize Alexandria, and the area around it including Arlington the home of Lee and of course the eventual cemetery.
As the Army of the Potomac was built massive fortifications were constructed on both sides of the river around the capitol, and many units spent almost their entire war manning the guns.
From the National park Service: "By 1865, the Defenses of Washington included 68 forts, supported by 93 detached batteries for field guns, 20 miles of rifle pits, and covered ways, wooden blockhouses at three key points, 32 miles of military roads, several stockaded bridgeheads, and four picket stations. Along the circumference of the 37-mile circle of fortifications were emplacements for a total of 1501 field and siege guns of which 807 guns and 98 mortars were in place."
They were only seriously tested once in late 1864 when troops under Jubal Early tested the outer defenses at Fort Stevens in an attempt to relieve pressure at Petersburg, however after a blooding they withdrew.
As for the rest of the river frankly it was hard to defend the entire length and was obviously not as evidenced by the Army of Northern Virginia as a whole crossing and going back on two separate occasions.