I think that at that point, the three partitioners of Poland were still quite happy with their informal agreement that Poland should be split and kept off the map, thus I doubt either of them would want to get more Polish lands. Also, here's the thing - at this point Russia was just out of the Crimean War, weakened and having made enemies with quite a few other powers. It wouldn't want to risk being the aggressor just a few years later, would it? In 1859, they weren't yet on very good terms with France - they were pretty isolated diplomatically speaking, actually. If Prussia decides that it is wary of Russian panslavic ambitions and backs up Austria with military force, then Russia is getting shafted. Or even if the Ottomans or the British join (or both). Essentially it'd be like Russia breaking the peace treaty established just a couple years before.
Russian-Austrian enmity had nothing to do with Galicia, as Russia wasn't Poland, and didn't claim any Polish-inhabited territories on an ethnic basis. Their rivalry had everything to do with the Crimean War and the subsequent fight for influence over the Balkans - with Russia trying to worm its way towards Constantinople and Austria trying to prop up the Ottomans and contain Russian influence.
Russia might be militarily able to snatch Galicia in 1866, but after the January Uprising, I doubt Russia would want even more rebellious Poles. (And once again, there would be no valid reason for a casus belli.) At any rate, with Galicia-Lodomeria being a core Austrian territory, it would lead to an even quicker German-Austrian reconciliation, with Austria doing its very best to reclaim Galicia with German backing, forcing Russia to defend its impoverished and ravaged hotbed of dissent against the German states and subsequently lose influence in Central Asia against Britain. Since Russia is an autocracy at this point, technically the incumbent monarch could go ahead anyway, but all of his advisors would be going "no no no wtf are you doing?" and it's rather unlikely altogether.
Basically, a Russian grab of Galicia would eliminate a source of territorial dispute between Russia and Austria
This doesn't even make sense anyway. It wouldn't eliminate the territorial dispute (if we assume that Russia ever seriously desired the area to begin with), it'd inflame it, with Austria now motivated by revanchism to reclaim the area. In fact, with little Russian interest in the region (no doubt they'd have taken the land if they had gotten into a major war with Austria for some other reason and won, e.g. if WWI had gone super-well for Russia, but I don't think they were actively looking to start a war for the land), the tension would only increase.
Ignoring everything else, Galicia-Lodomeria belonging to Russia would make Polish and Ukrainian nationalism more of a threat as their respective nationalist organizations would now be able to coordinate their efforts within a single country, with fewer borders to split them up.