I just completed browsing through my 1920 print copy of Andrew B. Booth's "Records of Louisiana Confederate Soldiers and Louisiana Confederate Commands" (State of Louisiana, New Orleans, 1920) and did not find Steven DeGruy listed even though there are several men surnamed DeGruy presented in Booth, including some variations in the spelling of the surname.
"Esteban" is the Spanish version of the name "Steven" but the only DeGruy with the initial "E" I could find was E. V. (aka Eugene) DeGruy who was a member of the Crescent Regiment, Louisiana Infantry. Eugene (or E. V.) died in the hospital at Corinth, Mississippi on May 7, 1862.
I next searched the National Park Service's online Index to the Compiled Military Service Records for the surname "DeGruy" and "Confederate" and "Louisiana". Still no "Steven" but there is a CMSR for a Private G. V. DeGruy who was enrolled in LeGardeur's Orleans Guard Battery. This name does not appear in Booth's "Records" but may represent someone who joined the battery in the fall of 1863. LeGardeurís Orleans Guard Battery is the unit referred to as "Beauregard's Battery from New Orleans" in the document you transcribed.
The Orleans Guard Battery was formed for Confederate service in March 1862 from volunteers then serving in the Orleans Guard Regiment, Louisiana Militia and sent to Corinth, Mississippi with the Orleans Guard Battalion. The battery was disbanded in the summer of 1862 and its members scattered to other units. A year later, General Bragg resurrected the Orleans Guard Battery and assigned it to the Army of Tennessee artillery reserves. General P. G. T. Beauregard commanded the Department of South Carolina, Georgia, and Florida from September 1862 until April 1864. The Orleans Guard Battery was transferred to Charleston, South Carolina in November 1863 at General Beauregard's request where it served in various fortifications around the city until Charleston was evacuated in February 1865. They fought in the Battle of Averasboro, North Carolina on March 15th and 16th in 1865 losing their artillery pieces to counter battery fire.
I would think the next steps in identifying this Louisiana soldier would be (1) to examine the CMSR for Private G. V. DeGruy of LeGardeur's Orleans Guard Battery, and (2) to check the 1860 Federal census for Orleans Parish. Keep in mind that, while the good folks in Columbia, South Carolina regarded him as a "full blooded Spaniard, quite intelligent, refined, and a gentleman", that doesn't tell us that he was "just off the boat" from Spain. Spain owned and governed Louisiana for 40 years prior to returning the colony to France in 1803. Napoleon turned around and promptly sold it to the United States. When title was transferred to the United States, Spanish administrators were still in charge of the colony. Spanish is as much part of the heritage of Louisiana as French even though the French often loathe to admit it!
One of my wife's family members was in LeGardeur's battery at Charleston and I have written a biographical sketch of him which incorporates an outline of the history of the Orleans Guard Battery.
Happy New Year!