Does the USA possess any capability to intercept supersonic cruise missiles (e.g. BrahMos)?
If YES, how?
If NOT, why?
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I have answered this very question in detail on a more relevant Aviation StackExchange.
I'll restate a few key points from my answer, and add a few relevant to your question.
This paper includes some math. This is for a point target being specifically defended, against an expected attack. There are multiple factors that can break the kill chain, and you can see how 50% can quickly turn into 5% if the missile is considerably more advanced.
Speed complicates interception in several ways. It gives less time to detect, identify, prepare to fire, and engage the target. It also multiplies the effect of error in tracking and targeting. Overall, the difficulty of interception scales approximately with speed squared. The paper above is for guns, but the math for missiles works in similar ways.
As can be seen from these sources, key factors that affect missile interception difficulty include:
The Brahmos missile represents the mid-upper tier of ~2010 Russian missile technology, augmented with Indian-developed software on Western hardware. It's been designed to defeat medium and large warships of modern navies, including the USN. Such ships are protected with at least 3 tiers of missile defense: SM-2 at a long range, ESSM within 30 nmi, and Phalanx CIWS at close range.
Given such targets, the Brahmos missile includes the widest range of electronic countermeasures among modern missiles, the highest speed, and is capable of various maneuvering profiles. So it's among the most difficult missiles to intercept.
Nothing would give a 100% interception rate. But to rely on a steady kill rate above 50%, one would have to employ a modern system with specialized anti-missile capability such as SM-3, MEADS, or S-400.
To bring the kill rate up further, to where 90% can be possible, one would have to step up to Integrated Air Defense Systems - a combination of long-range radars, combat air patrols, EAW/AWACS, fighters ready to scramble, and, then, layered long-range, medium-range and short-range SAM systems. 4 to 6 separate interception events, each with an independent chance to kill the target.
The technology would also have to be state of the art, such as S-500 or at least the newest Patriot and SM combinations, modern jets like the F-35, Typhoon or Su-30, with new missiles like Meteor or latest versions of R-77 or AMRAAM, plus modern local air defense systems.
The US has such defenses in place around its Carrier Battle Groups, and possibly around Washington DC and other critical military command centers. If fired upon, when an attack is specifically expected, one can expect most missiles to be deflected by decoys or intercepted by hard-kill measures. Some missiles, between 1 out of 10 and half, will still hit their targets. This is very approximate.
The "RIM-162 ESSM is designed to counter supersonic maneuvering anti-ship missiles."
The BrahMos is a version of the P-800 Oniks. It's top speed is around Mach 2.6. The ESSM's top speed is around Mach 4.
Details have generally not been released to press, but in 2013 Raytheon claimed to have successfully tested the ESSM against a "high-diving supersonic" threat.