We carried out detailed, self-consistent calculations for stars from 1 to 9 M_☉ over a wide range of metallicities, following the evolution and nucleosynthesis from the pre-main sequence to the asymptotic giant branch (AGB), in order to provide a self-consistent grid for evaluating stellar oxygen isotopic variations. These were calculated for first and second dredge-up, and for some masses also for third dredge-up and hot convective envelope burning on the AGB. We demonstrate that ^(16)O/^(17)O in red giant envelopes is primarily a function of the star's mass, while ^(16)O/^(18)O is primarily a function of the initial composition. Uncertainties in the ^(17)O-destruction rate have no effect on the ^(16)O/^(17)O ratio for stars from 1 to 2.5 M_☉, but do affect the ratios for higher masses: the stellar ^(16)O/^(17)O observations are consistent with the Landre et al. (1990) rates using ƒ = 0.2 for ^(17)O(p, y)^(18)F and ^(17)O(p, ɑ)^(14)N, and with the Caughlan & Fowler (1988) rates using ƒ ~ 1. The stellar ^(16)O/^(18)O observations require ƒ ~ 0 in the Caughlan & Fowler ^(18)O(p, ɑ)^(15)N rate. First dredge-up has the largest effect on the oxygen isotope ratios, decreasing ^(16)O/^(17)O significantly from the initial value and increasing ^(16)O/^(18)O slightly. Second and third dredge-up have only minor effects for solar metallicity stars. The absence of very low observed ^(16)O/^(18)O ratios is consistent with a major increase in the ^(18)O(ɑ, y)^(22)Ne rate over the Caughlan & Fowler (1988) value. Hot bottom burning in stars above about 5 M_☉ can cause a huge increase in ^(16)O/^(18)O (to ≳10^6), and possibly a significant decrease in ^(16)O/^(17)O; these are accompanied by a huge increase in ^7Li and a value of ^(12)C/^(13)C ≈ 3. The oxygen isotope ratios in the Al_2O_3 grains (Orgueil grain B, the Murchison 83-5 grain, and the new Bishunpur B39 grain) can be accounted for if they originated in stars that did NOT have the same initial ^(16)O/^(18)O ratio. Thus one cannot assume uniform isotope ratios, even for stars of nearly solar composition. The grains' ^(16)O/^(17)O ratios, together with the ^(26)Mg excesses that indicate grain formation in a ^(26)Al-rich environment, indicate that the Orgueil grain B and Murchison 83-5 grain originated in stars of roughly 1.5 M_☉, during third dredge-up on the AGB. The new Bishunpur B39 grain originated in a star of either 2 or of 4-7 M_☉.