You'd purt'near need a reactor to push that sort of current.
It does not require a nuke but it does require some significant power. It would be unlikely to retrofit onto our older destroyers but no problem for Zumwalt class. More ships are being designed that use generators and electric variable speed drives for propulsion, rather than direct engine to prop.
They have enough juice to charge up capacitors to deliver the burst of electric power for the rail gun.
...The ship's electricity comes from gas-powered turbines, which together generate up to 78 megawatts of power. Under normal conditions, the operating systems of the ship and the engines use only 20 of those megawatts, so the ship has 58 left over for weapons. Especially future weapons: Railguns and lasers, both long in development for the Navy, could find a home on board the Zumwalt, which has more than enough electricity to meet these weapons' intense energy demands....
Inside The Zumwalt Destroyer
https://www.popsci.com/article/technology/inside-zumwalt-destroyerNovember 21, 2013