Well I suppose they could. But as we've seen, there are plenty of 2nd level domains under .va, and in fact, many if not most of those, especially vatican.va, actually refer to the Holy See.

It's very common and understandable for folks to conflate the Holy See with the Vatican, but they are distinct entities with two different functions and purposes.

If we're discussing the administration/governance of the Church and the organs which make up the Roman Curia, then we're definitely looking at the Holy See. It's the Holy See, not the Vatican, which is a Permanent Observer to the U.N.

But again, if we're looking at any sort of physical presence, it's the Vatican City State. The actual territory that's an enclave and microstate is the Vatican. It's where people live, work, and worship. "The Vatican" is the entity selected for the ".va" ccTLD simply because of the way the rest of the world works. But only a few of those 2nd level domains can properly be considered more "Vatican" than "Holy See".