Could dark matter be made of particles that interact only gravitationally? Are current observations consistent with the hypothesis that dark matter is entirely composed of one or more types of particles that interact only gravitationally with each other and the usual matter?
What hope would there be for proving the existence of such particles by experimental means?
Or would our only hope be to find a sufficiently convincing theoretical framework that contains our currently best models, but also predicts the existence of such a "sterile" dark matter particle?
 A: It is completely possible for DM to have only gravitational interactions, and this is "nightmare scenario" is well-known by practitioners.
There are at least three hints against this possibility. First, there are a lot of astrophysical and cosmological anomalies (such as the galactic center excess) that could each be explained via some non-gravitational DM interaction. Second, for reasons of economy, we typically prefer theories of DM that can also explain outstanding problems in particle physics (such as the strong CP problem). Third, a theory of DM generally needs a way to explain how the DM was produced in the early universe, and most production mechanisms we know about (such as freeze-out) need non-gravitational interactions. However, it could still turn out that all those astrophysical anomalies have other explanations, that DM is completely unrelated to all problems in particle physics, and that DM was produced purely gravitationally.
If DM takes the form of black holes, they could interact only gravitationally and yet still be directly detectable, either through their Hawking radiation (for lighter ones) or through gravitational lensing (for heavier ones). But if DM takes the form of particles interacting only gravitationally, we're out of luck; we probably could never detect such particles directly.
In that case, the only way to test the DM hypothesis would be to do more precise measurements of its gravitational influence on galaxies and cosmology. This would be perfectly legitimate science, as there are plenty of objects in science that we've never observed "directly" but which still have great explanatory power. For example, astronomers living today have never and probably will never go to another star, but the hypothesis of stars is still universally accepted because they do a great job of explaining the points of light we see in the sky. Still, if DM ends up like that, it would be quite frustrating and disappointing, because in most theories DM is already right here, regularly passing through our planet.

Or would our only hope be to find a sufficiently convincing theoretical framework that contains our currently best models, but also predicts the existence of such a "sterile" dark matter particle?

In this case, particle theory would probably not say very much, because the models would be very unconstrained. It's just not that hard to add in extra particles that don't interact with the rest. If this happened, particle physicists would probably just step away to work on other problems.
