Yes, easily. We can actually take out most ICBMs using conventional explosives. I would be willing to bet we have a quick nuke drop warhead as a fallback if the conventional ones fail.
The trick is what kind of nuke to use and what altitude.
The nuke to use is an "enhanced" a.k.a "neutron bomb." The "enhanced" is a bit of misdirection. All fusion devices emit 99%+ of their energy as neutrons. To convert the neutrons into blast and heat and in general make a big explosion, you have to wrap them in a dense, neutron absorbing material like lead, polypropylene etc.
It's a myth that fusion nukes are highly destructive in space. With no atmosphere or ground to convert neutrons to heat and blast, you just get a rather large quick flashbulb for radiant effect. Space itself, even near earth space has so much volume that you have to be within something like 30km for a 1 megaton device to generate a killing pulse. (Intensity falls with the square of distance, remember.) On the surface, a 30km radius is massive, in space its a blip.
Neutrons don't kill other nukes by primarily heat, blast or frying the electronics. Instead, they transmute the isotopes within the enemy device, altering the critical ratios of those isotopes such that the device can never go critical. (Although, if close enough, the neutrons will cause heating in isotopes and blow it apart right there and some electronics can be fried by neutrons.)
So, the best point of intercept is above the atmosphere i.e. 90miles/140km or higher. The really important satellites are in geosynchronous orbit at 25,000miles/40,000km, so they're safe from any interception blast.
Intercepting above the atmosphere also prevents the blinding effects of ionized atmosphere noted by others. Even that may not matter as the enemy will be tracked by multiple sensors deployed on the ground, airborne and from high satellites, all of which will be transmitted to the interceptor which can otherwise fly blind. The ecological and other ground effects are minimal. With little blast or heat, there is little plasma and thus little EMP.
The real utility of an interceptor system is that it introduces immense uncertainty in calculating the success for an attack. Nobody really knows how all the factors in a nuclear attack will combine to produce what output. The interceptor system might substantially fail in a real attack or it might wipe out the attack completely. In the latter case, you've done nothing but p*ss off the targeted polity.
That uncertainty was a big part of the Reagan's Star Wars mojo back in the 80s that helped bring the Soviets down. The Soviets had long planned on being able to launch a devastating first strike and then absorb a much smaller counter-strike. The maybe-it-will-work,-maybe-it-won't Star Wars talk, threw that out the window.
Active defense is the new hotness at all levels. The Israels' are knocking individual artillery rounds out of the sky and pre-detonating RPGs. Interceptors in one form or another, and at all levels are here to stay.