The short answer is no—glaciers did not "dig out" New York Bay in the sense of gouging a basin like a spoon in soft earth. The actual origin is a fascinating interplay of river erosion and glacial isostasy (land movement).
New York Bay (comprising Upper and Lower Bay) is fundamentally a partially submerged river valley carved by the ancestral Hudson River. During the Last Glacial Maximum (~20,000 years ago), the massive Laurentide Ice Sheet covered the region. The immense weight of this ice depressed the Earth's crust downward (isostatic depression).
While the ice sheet didn't carve the bay itself, it enabled the river to cut deeper. Because the land was pushed down, the Hudson River's base level was lower, allowing it to erode a deeper channel further south than it does today. The river carved a valley through what is now the continental shelf.
As the climate warmed and the ice sheet melted (starting ~18,000 years ago), two things happened simultaneously:
The result: the deep river valley was flooded by the rising Atlantic, creating the expansive natural harbor we know today as New York Bay. The Verrazzano Narrows marks the approximate seaward end of that ancient river gorge.
So, glaciers didn't dig the bay—they depressed the land, letting the river dig deeper, then melted to flood the result.