The short answer is no. As we saw in Is It True That One Doctor Supposedly Debunked Andrew Wakefield's Autism Research?, no one doctor debunked Wakefield's research; it was a veritable international army of physicians, epidemiologists, and other researchers.
If there is one person who "debunked Andrew Wakefield's research", it is the UK investigative reporter, Brian Deer. Is he related to Rupert Murdoch?
I suppose it comes down to how you interpret "related". Is Deer's mother or father a related in any way to the Australian Murdoch family? It doesn't appear so, so NO.
Has Deer had a financial relationship with the Murdoch media empire? Well, yes. Deer's investigative reports were published by in the Sunday Times, which the Murdoch media group has owned since 1981. Did they "control" his reporting? No. Brian Deer's complete account of his relationship with the Murdoch media enterprise.
To review the claim
Dr. Wakefied's research never claimed vaccines cause Autism. His research showed that vaccines cause intestinal damage that led to leaky gut syndrome which allowed toxins to enter the bloodstream and brain, And THAT is what caused Autism. The doctor that supposedly debunked his research was funded by Merck, the manufacturer of the MMR and was related to Rupert Murdoch who obviously controls lots of the mainstream media. This man was also convicted of embezzling CDC funds. Over two dozen studies have since confirmed what Wakefields studies have shown.
The doctor that supposedly debunked his research was related to Rupert Murdoch.