Why do physicists assume that dark matter is weakly interacting? IceCube, XENON, etc, keep yielding negative results. If dark matter exists, it doesn't interact with baryonic matter at the energy ranges they can detect. The response is to build even bigger detectors to search for even fainter energy signatures.
Why? Is there evidence that dark matter is supposed to have weak interactions (instead of gravity-only)? Or is it just searching for your keys under the lamp post (i.e. it's the only possibility that we have a way to detect)?
 A: The short answer is that they don't assume that.
But among all the proposals that remain for what dark matter might be, weakly interacting stuff is the easiest to detect,1 so that is what is getting the money right now.2
And that is not unusual. The history of missing-mass/dark-matter is one of proposals being made and then ruled out one-by-one, in order of ease of accessibility. WIMPs are just the latest candidate to get top-billing. MACHOs were hot when I was in college but were largely disposed of in the nineties and naughties. Before that, decades were spent with ever improving telescopes in wider and wider bands just ruling out many of the ways that ordinary matter could be hiding in plain sight (gas and dust, mostly).
And there are additional possible candidates in the theoretical catalog. I think that sterile neutrinos and/or axions will be next up if WIMPs are convincingly ruled out.

1 There is a caveat here in the form of sterile neutrinos which are not "detected" exactly but can be deduced by finding the three-flavor mixing matrix to be non-unitary. This is a hot topic again because MiniBooNe has recently announced an improved analysis of a larger data set in which the low-energy excess remains and the $\theta_{13}$ efforts have paid off in a big way so we've close to being able to quantify the unitarity (or lack thereof) of the matrix with some precision.
2 WIMPs in a particular mass range also offer the possibility of explaining additional features of the universe which makes them attractive for a second reason.
A: It's not just the "look under the lamp post" effect. There's also the "WIMP miracle". A new heavy (i.e. about the mass of the top quark, the heaviest SM elementary particle) weakly interacting particle would have an annihilation cross section of about $10^{-26} \text{ cm}^3/\text{s}$. Very general thermodynamic principles predict that thermal production of dark matter in the early universe could only lead to the observed density of dark matter if the dark matter has a similar cross section. This similarity suggests that dark matter might consist of heavy WIMPS.
A: 
Note: this answer is similar to this one also written by me. That question is very closely related and an interesting read.


Why do physicists assume that dark matter is weakly interacting?

They don't assume that.
To answer your question, you need to understand how dark matter was hypothesised, so here is a summary:

Using supercomputers, physicists were simulating the Big Bang and the formation of the Universe, applying Einstein's theories of special and general Relativity and Quantum Mechanics, experimenting with different variables to try to arrive at a system similar to our world as it is currently.
As they experimented, they found that in the simulations generated by the supercomputers, the matter formed attracted each other too weakly; matter and gas were flung out too far during the Big Bang and could not "clump" together to form stars or planets.
They tried adding some "dark matter"; matter which did not interact with the strong nuclear, weak nuclear, and electromagnetic force, i.e. it only interacted with ordinary matter gravitationally. This "placeholder mass" solved the problem, and the digital model successfully evolved to the system of the cosmos we observer today.
The intriguing thing was that ~$85$% (!) of the universe had to be made up of this hypothesised "dark matter" so that it formed correctly.

Conclusion: the universe can't have existed without this mass made up of WIMPs (Weakly Interacting Massive Particles). So let's go look for it!
Dark matter is called dark because it is hard to detect, even though it is greatly abundant. Physicist don't assume that it is weakly interacting, it was named "dark" because it is so.
