Quality of canned apricots is seriously compromised by the loss of their flesh firmness during pasteurization. This work conducted a comprehensive characterization of the biochemical properties of raw apricots, as well as of the tissue softening during pasteurization using a multimethodology approach. Biochemical and textural results were analyzed to seek possible correlations, with the aim to elucidate which properties could be used as indicators in raw fruit for obtaining pasteurized products with better texture. There were large effects of cultivar, ripening stage, and intra-fruit zone on the biochemical and textural properties of apricots. The titratable acidity, amongst ethylene production, soluble solid content, and pectinolytic enzyme activities, was the biochemical property that correlated more robustly to the loss of texture. It had significant correlation coefficients of −0.9 and 0.5 with firmness of pasteurized fruit and loss of firmness, respectively. We propose titratable acidity to be used as indicator for raw material selection and/or to be modulated during apricot food processing.