The use of "fundamental" here refers to whether or not the right can be incorporated (i.e. applied to levels of government below the federal level), not whether the right exists--that's what this whole decision was about. Sotomayor joined in Breyer's dissent:
Rather, it directs its attention to this Court’s “incorporation” precedents and asks whether the Second Amendment right to private self-defense is “fundamental” so that it applies to the States through the Fourteenth Amendment. [...]
In my view, taking Heller as a given, the Fourteenth Amendment does not incorporate the Second Amendment right to keep and bear arms for purposes of private self-defense. Under this Court’s precedents, to incorporate the private self-defense right the majority must show that the right is, e.g., “fundamental to the American scheme of justice,” Duncan v. Louisiana, 391 U. S. 145, 149 (1968); see ibid., n. 14; see also ante, at 44 (plurality opinion)(finding that the right is “fundamental” and therefore incorporated). And this it fails to do.
Their answer was "no," which doesn't imply an argument for overturning Heller or that the right to bear arms doesn't exist (note Breyer specifically takes the Heller decision as a given there). There's no lie or even contradiction in your quotes.