Natural radioactivity represents one of the main backgrounds in the search for neutrinoless double beta decay. Within the NEXT physics program, the radioactivity- induced backgrounds are measured with the NEXT-White detector. Data from 37.9 days of low-background operations at the Laboratorio Subterráneo de Canfranc with xenon depleted in 136Xe are analyzed to derive a total background rate of (0.84±0.02) mHz above 1000 keV. The comparison of data samples with and without the use of the radon abatement system demonstrates that the contribution of airborne-Rn is negligible. A radiogenic background model is built upon the extensive radiopurity screening campaign conducted by the NEXT collaboration. A spectral fit to this model yields the specific contributions of 60Co, 40K, 214Bi and 208Tl to the total background rate, as well as their location in the detector volumes. The results are used to evaluate the impact of the radiogenic backgrounds in the double beta decay analyses, after the application of topological cuts that reduce the total rate to (0.25±0.01) mHz. Based on the best-fit background model, the NEXT-White median sensitivity to the two-neutrino double beta decay is found to be 3.5σ after 1 year of data taking. The background measurement in a Qββ±100 keV energy window validates the best-fit background model also for the neutrinoless double beta decay search with NEXT-100. Only one event is found, while the model expectation is (0.75±0.12) events. [Figure not available: see fulltext.].