Daystrom Institute

Data was artificial and autonomous. The Dr originally wasn't autonomous, it could be argued he's just part of the ship, but the holo emitter changed that. I'm amazed the Daystrom institute let him keep it, but since it's apparently his, and that makes him autonomous, I would argue he's just like Data (minus the permanent corporeality of course). I suppose there's a question about ownership given his origins as a Starfleet asset, but since he can be replaced with a copy of the original program, there's no real material loss in letting him leave the ship.

He would most certainly be assigned a rank if for no other reason than to ensure he (as an asset) is treated by crewmen with care and respect. It is the same approach in logic that militaries and police use with K-9 units. Dogs always outrank the handlers as a further mechanism to ensure proper treatment of the self-aware asset.

Even without an official rank, on Voyager he was still considered a Department Head and (more importantly) the CMO, which gave significant authority (even exceeding the Captain on certain medical matters), regardless of whether or not he was ever given any pips. The same thing would likely apply on subsequent postings.

If he ever had to be assigned a rank for clerical/administrative purposes, it would probably be the default required rank for a Starfleet CMO candidate for the class of ship he was serving on.

Who could stop Janeway from field commissioning him?

I thought he was the commander of the Atlantis expedition.

He doesn't need a rank to have authority over others. A duty station can give one authority, and he is the acting medical officer of the ship, so technically he can even order Janeway about.

Technically

I can’t imagine Janeway not at least trying to field commission him.

Cries in Ensign Kim

In Voyager, he's shown to have pips. In fact, switching him over to Command mode shows a deliberate animation of pips showing up on hid collar.

However, it is possible that this is something that only applied on the Voyager thanks to their excsptional circumstances, and regular Starfleet doesn't recognise it as a "proper" rank.

• Can a self-aware hologram hold rank or a non-com position in Starfleet?

Technically, yes, in practice, it would be a bit more complicated. A lot of the Federation still has issues around recognising the personhood of inorganics, and a good many of them would hold the early Voyager attitude of seeing him as a regular hologram/tool that the Voyager got too attached to, like the Enterprise did with their Data.

• If so, how would the Doctor attaib it?

The regular way, in theory. The ranking system technically doesn't change depending on what species you are, other than some minor twiddling to account for your species' characteristics.

It would be silly to expect a species who can't speak to give verbal commands, for example, or give them a crew who is not receptive to telepathy.

In practice, there's a lot more complications, like how the crew of the Sutherland nearly mutinied against Data because they believed him to be a dispassionate computer weighing lives at data points.