A graduate of the United States Military Academy in 1850, Grover was stationed in the western frontier before being transferred to help in defense preparations of Washington, D.C., at the outbreak of the Civil War.

He served as a brigade commander in Joseph Hooker's division of Samuel Heintzelman's the III Corps of the Army of the Potomac, in which role he won distinction at the Battle of Williamsburg and was brevetted lieutenant colonel in the regular army, and winning promotion to full colonel for gallantry at the Battle of Seven Pines.

He was transferred to the Department of the Gulf in December 1862 where he commanded a division in the XIX Corps during the Siege of Port Hudson.

Returning to the East in August 1864, he participated in the Third Battle of Winchester, Fisher's Hill, and Cedar Creek, where he was wounded and brevetted to the rank of major general of volunteers.

Before the war's end, Grover was brevetted to major general in the regular army on March 13, 1865, before returning to frontier and garrison duty with the U.S. Cavalry.