Shooting down BM's can be done - may already have been done by the Navy - but not with any standard AAW system installed today.
IIRC, the ship launched ABM's that have have intercepted the test ICBM launches, and intercepted the defunct spy satellite, were modifications of the SM3. And I thought the payload was just a straight kinetic intercept. I'd guess there's no point in a proximity warhead at those closing speeds. The target would outrace any blast wave or fragments other than a direct hit. IIRC the payload is just an IR camera with X-Y RCS thrusters. At that speed you could hit the warhead with a cotton ball, and it would destroy it.
The use of ballistic missiles in the anti-ship role (with conventional or nuclear warheads) has been discussed for decades. The biggest issue has always been how do you let the other side know that you are using it tactically rather than strategically? Are you trying to take down one ship with a conventional warhead, a task group with a nuke, or an entire base?
The same problem is one of the biggest obstacles to the conventional Trident program. Yes, we can have a weapons system that will take out a major facility within minutes of the go order, but how would China, Russian, et al react to an SLBM being launched and (likely) heading in their general direction?
Yes, that is the sticking point. Any ballistic activity, especially sub-orbital and inter-continental in distance has always been seen as a sign of the opening strike in wide scale strategic warfare, and not just limited tactical action. There's been several conventional warhead ICBM proposals, and other nifty things that deploy a UAV, even a rapid reaction force of Marines (!) etc. but they've all stalled for this very reason.
What does the POTUS do? Get on the red-phone and say: "
Hey, uhh.. yah it's ours... That's an ICBM, but it's not a nuke, m'kay?" I suspect we have A LOT of undeclared UAV capability, and that they involve high degrees of stealth, long loiter times, and perhaps even acting as carriers for stealth cruise missiles, guided GPS munitions etc. When you take pilot comfort and endurance out of the equation, what you can do in the sky opens up a great deal.