But apparently you could - or Pulaski could at least.
I'm guessing you are basing that off Pulaski's line "I'm not a Bridge Officer, ...."
I don't think she meant she wasn't qualified to command a watch, she meant that her job (and field of expertise) was in sickbay and not astrophysics (or whatever). Sort of an homage to "I'm a doctor, not a ...")
Now, I can't think of a single piece of evidence that Pulaski had any command training.
But I can't absolutely say that she didn't.
And I can see that someday being CO of a Starfleet Hospital might be one of her ambitions.
The bit between Troi and Crusher about "not needing the rank" bothered me quite a bit, too.
Remember that before Julian Bashir we'd not seen a Doctor who wasn't at least a full Lieutenant (I think they'd all been commanders, but maybe not.)
And the Nurses are all at least Ensigns.
How many Doctors and Nurses do you think the CMO of a Galaxy Class ship is in charge of?
I'm guessing at least six MDs, probably more like ten, and about a dozen Nurses. The floorplans show a veritable hospital, with multiple operating rooms, and that staffing is pretty similar to a Mobile Army Surgical Hospital (the real ones, though the fictional ones too).
And the CO of that unit would be a Lieutenant Colonel, which would be a Commander in Starfleet.
To be CMO on a Galaxy Class, yes you
do need to be a Commander.