Use separation papers to identify a US soldier's WWII service in Northern France GO 46 WD 45

Knowing he was with the 44th Division tells us something and gives clues for getting more information. Wikipedia has a page for the 44th Infantry Division, which was originally known simply as the "44th Division" and was made up of East Coast National Guard units. Hopefully that's consistent with the area he came from?

The Wikipedia page reckons that the division landed at Cherbourg in September 1944, spent a month training, fought in the Vosges Mountains and took part in the liberation of Strasbourg. It then moved on to Ratzwiller and Fort Simserhof, repelled German counter-attacks, crossed the Rhine on 26th March, took Mannheim, and then came out of the line for three weeks for training. Returning to combat in late April, it crossed the Danube and headed for Austria, taking the surrender of Wernher Von Braun and his group of V-2 rocket scientists along the way. It reached Imst, Austria, before the end of the war in Europe, did a brief period of occupation duty and then returned to the US for re-training. The sudden end of the war in the Pacific meant that the 44th never deployed there.

The commanders in Europe were Robert L. Spragins and William F. Dean and their Wikipedia pages give more detail, and add inconsistencies to the story.

Wikipedia also gives us links to a site specifically about the 44th, and to a site at the US Army Centre for Military History, which has quite a lot of administrative detail.

Since your father was a member of the 44th Division Artillery, he would be part of the 44th Infantry Division, you can follow its combat activities after it landed in Cherbourg, France on 15 September 1944. Since you indicate your father had both Northern France (GO 46) and Central Europe(GO 48) campaign designations, it seems probable his unit followed the progress listed for the division:

The 44th Infantry Division landed in France via Cherbourg, 15 September 1944, and trained for a month before entering combat, 18 October 1944, when it relieved the 79th Division in the vicinity of Foret de Parroy, east of Lunéville, France, to take part in the Seventh Army drive to secure several passes in the Vosges Mountains. Within 6 days, the division was hit by a heavy German counterattack, 25–26 October. The attack was repulsed and the 44th continued its active defense. On 13 November 1944, it jumped off in an attack northeast, forcing a passage through the Vosges Mountains east of Leintrey to Dossenheim, took Avricourt, 17 November, and pushed on to liberate Strasbourg, along with the 2d French Armored Division. After regrouping, the division returned to the attack, taking Ratzwiller and entering the Ensemble de Bitche in the Maginot Line. Fort Simserhof fell 19 December. Displacing to defensive positions east of Sarreguemines, 21–23 December, the 44th threw back three attempted crossings by the enemy of the Blies River.

An aggressive defense of the Sarreguemines area was continued throughout February 1945 and most of March. Moving across the Rhine at Worms, 26 March, in the wake of the 3d Division, the 44th relieved the 3d, 26–27 March, and crossed the Neckar River to attack and capture Mannheim, 28–29 March. Shifting to the west bank of the Main, the division crossed that river at Gross-Auheim in early April, and engaged in a 3-week training period. Attacking 18 April, after the 10th Armored Division, the 44th took Ehingen, 23 April, crossed the Danube, and attacking southeast, took Füssen, Berg, and Wertach, in a drive on Imst, Austria. On 2 May, a group of V-2 rocket scientists that included Wernher von Braun surrendered to the 44th.[9] Pursuing the disintegrating enemy through Fern Pass[10] and into the Inn River valley, the 44th set up its CP at Imst on 4 May. After a short period of occupation duty, the division returned to the United States in July 1945 for retraining prior to redeployment, but the end of the Pacific war resulted in inactivation in November 1945 at Camp Chaffee, AR.

Another history of the 44th Division can be read in the Congressional Record.

Since his actual unit listing was Headquarters Battery, and his position of Supply NCO 821 indicates he was a non-commissioned officer,and a Quartermaster Supply Technician, he was probably not right on the lines, but keeping the artillery units attached to the 44th supplied with fuel and ammunition probably kept him fairly close to the front, perhaps trailing just behind the progression described above.