The 337th Infantry REGIMENT (not Division) was a part of the 85th Infantry Division (not the 88th ID).
The campaigns were the same for the 85ID and 88ID.
Casualties for the 85ID:
1,516 KIA 6,314 WIA 175 DFW, for a casuality rate in excess of 50%.
In addition, the 85ID left the line around 23 Nov 44.
It returned to combat 9-17 Jan 45, in the relief of the 1st British Division, in the area of Monte Grande.
One would assume that your father was KIA in the M. Grande operation.
Going back a little further in the division's military history, the 337th IR is listed as having departed Hampton Roads and Newport News, PE, 1 Jan 44, arriving at Casablanca, French Moroco, 9 Jan 44, where it trained for the subsequent engagements in Italy, arriving at Naples, Italy, 15-27 Mar 44.
It went into the line west of Minturno, 10 Apr 44.
If you know the periods of his overseas tenure, I may be able to identify his specific activities. OK ?