I don't think it takes an anthropologist to analyse whether Rosling's numbers are verifiable or not. He certainly does not provide evidence in the TED talk that they are correct, in that sense they clearly are not verifiable from the talk. If that's what your friend means then fair enough. A TED talk is not an academic publication.
I'm not aware that the field of anthropology rejects in principle the notion of child mortality being a real thing. So Rosling's comparisons could be checked against other sources. It would not be appropriate to cite Rosling in a paper if you needed a source for the relative child mortality rates of those pairs of countries. You would go to WHO and/or national medical reporting, and pay close attention to their methodologies. In that sense they can be checked, and either verified or falsified. Will your friend give you long odds that Rosling has those pairs of countries in the right order according to WHO or his preferred source? Thought not ;-)
HeRosling also hasn't really demonstrated anything about the state of knowledge of Swedish students, although the insinuation is that he has. He (serious-jokingly) says that there's a role for him to teach them something. His small trial is sufficient to support his small claim. His methodology is simple and obvious enough that the test is repeatable with other groups. There's no problem of fundamental science here. I don't know (and I don't think it matters) whether he's ever published on that particular result, but it's probably never appropriate to cite because it's such a limited and specific claim of no general interest. And might be cherry-picked.
It's all on the credibility ofTED talks are as credible as the individual speaker. The fact that TED has "chosen" them should lend no authority at all but probably, unfortunately, does. As for the credibility of TED as a forum, I don't recall what wit here in the UK observed that the country has very many people who would turn down an honour in principle, but tragically are never offered one. I suspect the same may be true of TED -- there are various reasons you might not want to do it, among them that the content of the talks tends to be over-trusted by people who enjoy TED and its speakers. These reasons are not universally applicable, and in any case apply most strongly to those who won't be invited.