Seems fair to me. Objects of science are studied empirically, works of art are created by man.
Yup, sounds about right. If any gods actually existed, of course, they could indeed be studied, and therefore would become objects of science...
Not as simple as all that, I'm afraid. Bear with me for a moment as I open up a big can of world building.
First a disclaimer. My actual view is that Gods are a mental manifestation of part of how people think. That is I regard them as instances of certain kinds of mental processes that have no independent existence from the human mind. This doesn't mean that I think they're imaginary, rather that they are ways of thinking that can be (but don't have to be) given form in mind. From this point of view some people have an easier time with those forms manifest in mind (with external reminders such as icons), and some people have an easier time with them as unmanifest processes.
Anyway, on to the world building. It's perfectly possible to conceive of worlds that look exactly like ours that have one or more external deities that can affect the world in such a way as to not be subject to scientific analysis.
The simplest is to suppose a god that can inspire particular thoughts in people or push them toward certain decisions. Because the actions of these gods would work with standard mental processes (ideation and inspiration) their actions would not be discernible.
More complex is to imagine gods that can choose among outcomes of events, essentially gods of luck. If such gods favored their followers then the fact those people were unusually lucky could be determined by statistical analysis. But if the god has a more Illuminati-like attitude and is playing some long game for an outcome we can't see then it would be impossible to find which events had undergone such divine intervention.
A fairly extreme example is a god existing outside of spacetime who creates the universe, looks at it, decides it isn't quite what the god wishes, destroys it, and creates it again. This god is from our perspective omnipotent and omniscient, but does not intervene in the universe. From our POV in spacetime we do not realize that the whole of our existence, past, future, etc is a transitory flash in the thoughts and actions of this god.