Upvote:2
US Army: Unit Records 1917-1950 from the Eisenhower Presidential Library lists, in box 745 on page 53, the 1186th Engineer Combat Group. Thus, from your note, that would appear to be your father's regiment, in the medical detachment.
An Engineer Combat Group was the engineering equivalent of a regiment, though varying more widely in composition depending on circumstances, from as few as about 3 companies to as many as a number of battalions.
The New York Historical Society in its WWII & NYC exhibition includes the story of New Yorker Nick Tanis, Technical Sergeant; and notes that "After D-day the regiment’s title was changed to the 1186th Engineer Combat Group":
Tanis was drafted and inducted on April 21, 1942 at Fort Jay. Tanis was initially part of Headquarters Company, 2nd Battalion of the 531st Engineer Shore Regiment ... [and] landed at D-day at H-Hour plus 2 (8:30am). ... After D-day the regiment’s title was changed to the 1186th Engineer Combat Group. Tanis was in the H&S Company of the 3052nd Engineer Combat Battalion, one of the three battalions in the 1186. ...
The Independent Record's article of Jan 28, 2005, on Vets Remembering the Battle of the Bulge notes:
Bill Hunt was in communications with the 3053rd Engineer Combat Battalion and the 1186 Engineer Combat Group. They were also used as infantry to assist the Ninth Army before transferring to assist the British Army under Montgomery.
The 531st Engineer Shore Regiment was attached to the 1st Engineer Special Brigade (also here) for the war, including the Normandy landings.
Upvote:3
The regiment number may not be listed as such, but it could be hidden in the jargon, so don't get discouraged if it's not immediately obvious.
The battles and campaigns you've mentioned like Normandy and Northern France were indeed under General Patton’s Third Army, so that might be a clue there.
Here's a suggestion. Have you tried getting in touch with the National Personnel Records Center (NPRC)? They could potentially provide more detailed information about your father's service. Be aware, though, that you'd need to be the next of kin or have the appropriate permissions to access these records.
And finally, consider looking it up using e-resources. Just note that the level of detail in a military service info confirmation can vary depending on the source and the extent of the check. In most cases, the person's service number, dates of service, and discharge status may have to be confirmed.