It depends on what you mean by "broadcast to the whole world".
According to the annotated transcript (h/t to Organic Marble for finding it) John Young said it into a "hot mike", i.e. with the air-to-ground communications loop open:
[In the following, John doesn't realize he still has a hot mike. Charlie is only faintly audible through John's mike and the following undoubtedly contains transcription errors.]
128:50:37 Young: I have the farts, again. I got them again, Charlie. I don't know what the hell gives them to me. Certainly not...I think it's acid stomach. I really do.
So it was broadcast over a radio link pointed toward Earth, and anyone on that side of the planet with the proper equipment could have picked it up.
However, communications between the spacecraft and mission control were not routinely rebroadcast. Mission control preferred to keep close control over communications with the public, and 12 days of cryptic acronyms would bore even the most avid space fan. To avoid situations exactly like this one where the astronauts could cause embarrassment, audio (or video) from the spacecraft would only be rebroadcast on scheduled occasions (and even by the time of Apollo 13, there was little demand for such broadcasts; I don't know if any were scheduled for 16 apart from lunar surface EVAs).
"I have the farts" could have caused a minor scandal if rebroadcast, but it's nothing compared to the language Young uses a few seconds later:
128:50:45 Young: (Laughing) I mean, I haven't eaten this much citrus fruit in 20 years! And I'll tell you one thing, in another 12 fucking days, I ain't never eating any more. And if they offer to sup(plement) me potassium with my breakfast, I'm going to throw up! (Pause) I like an occasional orange. Really do. (Laughs) But I'll be durned if I'm going to be buried in oranges.
After Gene Cernan's "son-of-a-bitch" on Apollo 10*, the entire astronaut corps was reminded to watch their language while on mission, so it's extra funny here that Young softened "I'll be damned" while letting the f-bomb go completely unfiltered.
Further supporting the idea that this incident wasn't rebroadcast at the time, it certainly doesn't appear to have harmed Young's career any; he went on to command two early shuttle missions and served as Chief of the Astronaut Office from 1974 to 1987.
Amusingly, it's almost four minutes from "I have the farts" until CAPCOM Tony England advises the crew that they're broadcasting.
128:53:58 England: Okay, John. We have a hot mike.
128:54:07 Young: How long have we had that?
128:54:10 England: Okay. It's been on through the debriefing. (Pause)
Amateur radio operators did pick up some transmissions from Apollo missions on some occasions; this article describes listening in on the VHF transmissions between the CSM and LM. I don't know enough about radio to know how specialized the gear needed to listen in on the S-band transmissions between the spacecraft and mission control would be, but apparently at least some operators were able to do it during Apollo 15 and 16. It would be highly unlikely that, out of nearly 300 hours of mission time, a ham would happen to be listening in at the moment John Young was discussing his flatulence.
Young is blaming orange juice giving him an acid stomach for the gas at this point; a half hour earlier we have this:
128:18:33 England: [...] And your biomed looks great down here. Just keep up the orange juice. (Perhaps) push on it a little bit there and everything will be fine.
128:18:58 Young: (Amused) Push on the orange juice and everything will be fine?
128:19:02 England: Yeah, push on the orange juice. Rog.
128:19:04 Young: I'm going to turn into a citrus product is what I'm gonna do.
128:19:09 England: Oh, well; it's good for you, John.
[The Apollo 15 crew experienced heart beat irregularities which, post-flight, were determined to have been caused by a potassium deficiency. In an effort to eliminate the problem, the Apollo 16 crew was given large quantities of potassium-fortified fruit drinks.]
From here, Young goes on at some length about his acid stomach, and England promises he can have an antacid tablet after the mission. I seem to recall that another contributing factor to flatulence was that the drinking water coming from the hydrogen-oxygen fuel cells tended to have hydrogen bubbles in it, which had to leave the astronaut's bodies one way or another, but the drinking water on the LM was not fuel cell water.
* Actually, Cernan let "son-of-a-bitch" fly at least 5 times in a two hour span on that mission!