Not the pilot episode, but the feature film that introduced the concept, the sub, and most of the major characters who would later appear in the TV series.
Interestingly, the film appeared in 1961, which was before the test cited in TFA. And more interesting still, the "fix" in the film was exploding a nuclear device (launched on a missile from the Seaview) to "blow out" the fire in the belts. I'm guessing that there had been some discussion of the test in years prior, and Irwin Allen spun a script around it.
I was only nine at the time, but I remember thinking that the whole concept of the Van Allen belts "catching fire" was absurd because I knew that the belts were outside the atmosphere, and fire requires oxygen. It was the first time I can recall thinking that the science in SF movies was BS. It wasn't the last.