This is a myth. The increased effectiveness of the F-4 during the Vietnam War is attributed to the Navy's TOPGUN program. From "The Transformation of American Air Power" by Benjamin S. Lambeth (48):

...seemingly in counterflow to the teachings of the Vietnam experience, the Air Force all but discontinued dissimilar air combat training (DACT) after the end of Rolling Thunder in 1968, just as the Navy was activating its Topgun effort. Not surprisingly, when the air war over North Vietnam resumed in 1972, the Navy's kill ratio rose dramatically, while that of the Air Force declined. Before the establishment of the Topgun program, the overall Navy kill ratio was 3.7 to 1. Afterwards, it shot up for a time to 13 to 1, leading naval aviators to speak of the Vietnam air war as having been conducted in two phases - "before Topgun and after Topgun."

and

From 1965 through 1968, the Air Force and navy had a roughly equal kill ratio (2.25 to 1 for the Air Force and 2.42 to 1 for the Navy). In 1970-1973, the Navy's ratio rose to 13 to 1, while that of the Air Force declined to 1.92 to 1.

It is important to note that the United States Navy did not operate the F-4E, the variant of the Phantom with an internal gun.

Furthermore, if we look at kill statistics for only the United States Air Force during the Vietnam War [source], F-4C's and F-4D's killed 72 MiG's with missiles but only 10.5 MiG's with their external gunpods. Meanwhile, F-4E's killed 15 MiG's with missiles and 5 MiG's with their internal cannons. Thus, while USAF F-4E's downed a larger proportion of enemy fighters with guns than USAF F-4C's and F-4D's, the majority of their kills were still made with missiles. And, as previously stated, the performance of Phantom's in the USAF lagged behind their internal-gunless counterparts in the USN.