We prove a Nakai-Moishezon criterion for adelic R-Cartier divisors, which is an arithmetic analogue of a theorem of Campana and Peternell. Our main result answers a question of Burgos Gil, Philippon, Moriwaki and Sombra. We deduce it from the case of adelic Cartier divisors (due to Zhang) by continuity arguments and reductions involving a generalization of Zhang's theorem on successive minima.