I didn't say Obama's inaction in Syria exacerbated the situation there, I said he did nothing to halt the flow of refugees out of the country, and that that harmed US-European relations. At some point in the course of the war, he could have set up safe zones that would have allowed civilians in Syria to remain safely in Syria without having to leave for neighboring countries, or, in some cases, European countries. One of the losers of American inaction on the refugee problem has been Europe, and that has, in turn, harmed the US: the flight of refugees to Europe has caused all sorts of social problems in Europe, and may even have contributed to the rise of Islamaphobic sentiment and the empowerment of right-wing parties who are opposed to NATO and want to reorient Europe away from the US and towards Russia.
Eh, I'm not quite so accusatory with this as you are (surprise!). To my mind, the larger, systemic changes have less to do with America being self-serving, and more to do with America having to reposition itself as China rises, and a new multipolar order emerges. (Obviously, though, it isn't reducible to that.) A direct consequence of Obama's very reasonable pivot to Asia was fewer soldiers stationed in Europe and a perception among Europeans that America was de-prioritizing Europe, after it had been central to American foreign policy since end of WWII. That left some of Europe disenchanted with the president, concerned about US foreign policy in the future and worried about their own safety. Europeans can complain all they want about American vulgarity, and condemn American actions abroad as moral outrages (G-d knows, they aren't always wrong -- but they aren't always right either, and very often their arguments are self-serving and hypocritical), but they also rely on American military power for their self-defense in 2017. So among other things, one of the more damaging things to American-European relations has been that America's global strategy has involved repositioning its limited resources to another place in the world where it's more useful, and more necessary -- and they don't like that we're leaving, and we don't like, as Obama said, that they're "free riding".
And, whatever -- Europeans can complain about US-Saudi relations all they want, but their governments are also allied with Saudi Arabia. They are no less "enablers" of Saudi Arabia than the United States is.