I wouldn't think so. I guess if you were incredibly close to the intercept point (and not necessarily the launch point) already, they might have enough propellent to chase something before it accelerates up, up, and away. You also run into a problem because the Phoenix is relying on aerodynamic surfaces to steer, and that caps your effective altitude and minimum velocity. So you'd have to be real close, I would think to the intercept point.
There would be serious issues with fusing, and guidance would be a chore because it could not simply chase a missile to be successful, it would have to compute a likely intercept point for a rapidly accelerating object.
The F-14 was studied as a carrier for a HTK missile as part of the Boost Phase Intercept program in the mid-90's (along with the F-15 and ... B-52 which would have carried several). I believe it was called Peregrine, but when attempting to confirm, I run into results for the more recent and smaller missile by the same name. Later there was talk of a PAC-3 derived missile called ALHK for BPI from airborne platforms, and NCADE which was a modified AMRAAM. Same limitations applied as with the Phoenix, perhaps more so in the case of NCADE.
Under reported limitation: the time to detect the launch, acquire the target, share data with launch platform, all happen during the ballistic missile's most vulnerable period. If one cannot see it on radar until it reaches 20,000' because of a radar horizon, for example, the ballistic missile is already half the way to an altitude where one's intercepting missile begins to lose some efficiency from it's surfaces, perhaps more so because heading "uphill" as it were is going to impact the speed, and thus effectiveness, of the fins or other aerodynamic surfaces.
And hitting one late possibly leaves it with enough energy to come down somewhere you really do not want it to (western Europe, Japan, US, Canada, etc).
So probably no useful concept of operations for the Phoenix in this role. You really need a missile that accelerates much faster, and probably one that has a longer range. Which translates to weight, size, and cost, though new propellants will obviously help the first two factors. Obviously, if you could somehow get closer to hostile airspace, that might also help you get closer to an intercept point which increases your chances of success...