Measurements of inclusive two-particle angular correlations in proton-proton collisions at centre-of-mass energies of 900 GeV and 7 TeV are presented. The events were collected with the ATLAS detector at the LHC, using a single-arm minimum-bias trigger, during 2009 and 2010. Correlations are measured for charged particles in the kinematic range defined by pT > 100 MeV and |eta| < 2.5. In total, integrated luminosities of 7 1/ub and 190 1/ub are analysed for 900 GeV and 7 TeV data, respectively. At 900 GeV only events with a charged particle multiplicity n_ch >= 2 are analysed whereas at 7 TeV, a second phase-space region of n_ch >= 20, with a suppressed contribution from diffractive events, is also explored. Data is corrected using a novel approach in which the detector effects are applied repeatedly to the observable distribution and then extrapolated to a detector effect of zero. A complex structure in pseudorapidity and azimuth is observed for the correlation function at both collision energies. Projections of the two-dimensional correlation distributions are compared to the Monte Carlo generators Pythia 8 and Herwig++ as well as the AMBT2B, DW and Perugia 2011 tunes of Pythia 6. The strength of the correlations seen in the data is not reproduced by any of the models.