Your DM is wrong
tl;dr/mathless: The Eye is Invisible and under an inch in diameter; much too small for an enemy firing blindly to even hope to hit, and too small to make any noticeable noise without smacking into an object at speeds much higher than it can attain normally.
Also, the RAI and RAW is that it can't be attacked, because it doesn't have stats or Hitpoints. The scenario your DM is proposing is extremely far-fetched, bordering on ludicrous.
For the purposes of being complete, I will make this much harder than it needs to be
Arcane Eye creates a "Magical Floating Eye" that hovers above the ground and can see 30 Feet in any direction, even in the dark. It is not said whether it is intangible or not, but text within the spell would suggest that it is, in fact, tangible.
A solid barrier blocks the eye's Movement, but the eye can pass through an opening as small as 1 inch in diameter.
The Eye has enough of a physical presence to not be able to move through a solid object, or any hole smaller than one inch.
it's still really tiny. Creatures can, in-general, fit through a hole of their size category, or one lower if they squeeze (PHB 192). The Eye does not have to squeeze to fit through that inch (inferred from the fact that it loses no additional movement when doing so, unlike squeezing, which uses twice movement).
That means the Eye, which cannot squeeze, is just under one inch in diameter (it "creates an eye", and eyes are roughly-spherical), because it cannot pass through any hole smaller than an inch in diameter, but can pass through any size larger hole. This would give it an approximate volume of:
$$4/3π * 0.5^3 ≈ 0.52cu.in$$
By comparison, the smallest Size Category in D&D 5e, Tiny, occupies a space of ~2.5x2.5ft (or 30x30in) cube, meaning it has a volume of:
$$30^3 = 27,000 cu. in$$
Your Arcane Eye is ~2*10^(-5) times as large as the Tiny size category
This absolutely miniscule size, coupled with the fact that it is invisible and floating, would mean that even if a creature managed to detect it somehow without relying on sight, the eye is almost impossible to hit. That's not "Attack at Disadvantage," difficulty that's "Attack at Disadvantage, only a Natural 20 on the lowest die Hits" difficulty.
It also means that it is incredibly improbable that a creature will somehow blunder into the eye without your eye being able to dodge it (since all actions happen simultaneously). The eye can go through an opening of one inch, and there are usually several inches between most humanoid's legs (Of course, there are always Gelatinous Cubes.)
And this is even if it were possible to attack it, since it has no stats, no AC, and no Hitpoints.