The claim comes directly from the organizers and seems to be nothing more than self-promotion from some press releases being carelessly repeated over and over. Exactly what it means depends on how you chose to interpret “watched the Tour de France” but it's certainly not the case that 3.5 billion different people watched the live broadcast at some point during the race.
Since ASO apparently does not provide any details regarding their methodology, I am not sure it makes much sense to speculate on how they come up with such a high number but there are at least a couple of plausible solutions:
- They consider each stage as a separate event. In the current format, that's 21 days of broadcast. Many people just tune in occasionally but cycling fans do watch most of them so would be counted several time. It's not entirely absurd (I think that's how you would count stadium attendance and the like, maybe it's standard in this industry) but it seems a little misleading when presented without explanation.
- They consider potential audience, as suggested by this article from Le Monde. So if some Chinese broadcaster paid for the right to show a few excerpts from the race, you add 1.3+ billion people. Not entirely absurd either but that does not translate to 3.5 billion people actually watching and cannot be compared to the figures you found for other events.
It's also easy to show conclusively that there aren't 3.5 billion different people watching the race. The world's population is a little over 7 billion people. As you noted, it would mean that every other person on earth watched the Tour de France, which is not plausible. Even assuming higher ratings in Western Europe and maybe the US, you would still need a significant proportion of the population in Africa, South America, China or India to reach that number.
Actual measures in the core audiences categorically rule that out. Daam Van Reeth studied data from Flanders, probably the region in the world where interest in this sport is highest. The highest peak audience ever was 1.4 M people, which is slightly above 20% of the population and the average audience is 8%. Impressive but a far cry from the 50% implied by the 3.5 billions figure.
In France, the end of the 2013 Alpe d'Huez stage was watched by 6.2 M people, with an average of 4 M people for other stages. It's a lot but still less than 10% of the French population watching any given stage. Another article based on the broadcaster's own figures has slightly different numbers (in the same order of magnitude) and also reveals that 24 M people watched at least one hour over the three weeks that the race lasts (perhaps the most straightforward definition of “watching the Tour de France”). This amounts to 36% of the French population. That's getting closer but still not 50% and that's for France itself.