Not by the hair on his chinny-chin chin. While surface-to-air missile (SAM) systems have been able to shoot down incoming missiles for decades, these rely on tremendous tracking systems, with large missiles. Ships, too, have radar-guided gun pods capable of destroying incoming missiles, but these are also large, with giant sensors and three times the ammo capacity of today’s fighters. These solutions would be inappropriate for inclusion in any fighter design.
What the fighter does have is wholly unsuited to shooting down an incoming missile. The radar cross section of the inbound is too small for the tiny sensor mounted in the pointy nose and by the time the SAM is within range, it’s likely no longer boosting, making it a poor target for a heat-seeking missile (some of which can be fired from high angles off boresight). The inbound missile is coming at such a speed that getting the nose around and taking bead for a gun shot is like like playing “chicken” with an oncoming train.
What might be feasible in the foreseeable future is some directed energy weapon capable of setting off the warhead prior to impact. This would take quite a lot of energy and the designers would need a strong reason to believe that several other aspects of the fighter should suffer for this defensive capability. It’s an unlikely trade-off. What is far more likely is continued advances in decoys and avoidance.