Background: The CNO cycle powers core H burning, in stars with M>1.2M, and shell H burning, during the advanced evolutionary phases. Therefore, the uncertainties affecting the reaction rates of proton captures on C, N, and O isotopes limit our understanding of stellar evolution and nucleosynthesis. Purpose: We aim to develop a general and self-consistent tool for the calculation of nuclear reaction rates and their uncertainties, starting from available experimental data. As a longer term plan, we intend to use this tool to revise the proton-capture reactions of the CNO cycle for which new experimental data are or will be available in the next future. Method: The general procedure consists of R-matrix cross section calculations based on available measurements of the relevant nuclear parameters (energies, widths, strengths of known resonances, interference patterns, etc.), coupled to a Monte Carlo procedure to evaluate the global reaction rate error. Results: A first application of this method to O17(p,γ)F18 and O17(p,α)N14, the reactions that determine the O17 destruction in stellar interiors where the CNO cycle is active, is presented. These two reactions also allow us to test the multichannel and multilevel capabilities of the R-matrix method. In the temperature range of hydrostatic H burning (T<100 MK), we confirm that the median rates are up to a factor of 2 higher than those suggested in reaction rate libraries commonly used in stellar model calculations. In this temperature range, the O17 destruction mainly proceeds through the O17(p,α)N14 channel, whose rate is known within ±20% (95% C.L.). Conclusions: Based on current stellar models of red giant stars, we show that this uncertainty produces a 10% variation on the predicted O16/O17 abundance ratio. Other uncertainties, such as those affecting the O17 production rate, i.e., the O16(p,γ)F17 reaction, have a stronger impact on this theoretical prediction, a fact that motivates further experimental investigations of the O17 production channel.