The USN would be the problem, as AWG-9/AIM-54 combination would not fit on F-4. But how about Lockheed "F-3" Viking based fleet defense fighter a la Douglas Missileer? As secondary role it could carry tons of standoff weapons for ASuW work or various bombs for colonial police actions. A "F-3" could stay on CAP practically for ever, and it could even mount a galley and toilet a la Su-34.
While the S-3 or A-6 airframes could handle the payload of the radar and 6xAIM-54, they are subsonic. The Missileer would have been more practical against the subsonic Tu-95 and T-16; against the Tu-22M threat that's an issue.
The carrier air defence team aims to get a CAP element to shoot down your SNA Tu-95D before it can paint the carrier and pass on its location. Failing that, it aims to get the CAP to shoot down the regiment of Tu-22M bombers before they launch their Kh-22 Raduga/AS-4 Kitchen. Failing that, it aims to get the CAP to shoot down the missiles in flight, followed by the group area air defence ship (usually one or two CG or CGN) to down the missiles, or to get the inner air defence ring of DDGs to down the missiles, or to get the close escort frigate to down the missiles, or to get the carrier's array of Sea Sparrows and Phalanx to shoot down the missiles, or to evaporate in a nuclear fireball (note - the last option is considered to be a suboptimal outcome to the exercise).
If your CAP flights are not on the threat axis (and some of them won't be) , then they need to dash to get to within the ~100+nm range of the AIM-54 to be able to engage (and those Tupolevs may be hooning along at 1100kts themselves). That's where a 1200+kt afterburning F-14 is vital over a 450kt MPA or attack aircraft; to sprint the 50nm to get into range before the MPA gets its transmission or datalink off, or the bombers get within range of the carrier. When it takes an extra 3 minutes to get into the missile envelope, that's the margin that makes the difference between trying to shoot down Mach 1.5 bombers before they launch, and Mach 4.5 missiles afterwards. Which means you need more CAP elements to maintain coverage of the area (since they can't do a supersonic sprint to get into range, they have to already be in range at detection). Which increases wear on machines and crew, burns out parts and burns up avgas, and is basically harder than relying on two pairs of F-14 for the same coverage.
Of course, if fixing the F-111 to be a navalised fleet defender was easy, it probably would have been done OTL...