I had the impression that Obama’s remarks to Mitt Romney - "the 1980s are now calling to ask for their foreign policy back, because, the Cold War’s been over for 20 years" - were a bit earlier than 2010, when he wrote those words, or 2012, when Obama use them to undermine Mitt Romney’s campaign. In hindsight, Obama put his foot in it; Russia would take chunks of Ukraine during Obama’s second term in office, and then go on to attempt to seize the rest of Ukraine after Biden took office. Romney was right, at least to a very great extent.
More seriously, the Russian Georgian war was fought in August 2008. It’s at least possible that a more successful war would turn Obama into an anti-Russian hawk, or grant victory to John McCain in the November 2008 US election. This could have interesting knock-on effects. For one, Obama’s decision to drawdown US combat forces in Iraq severely weakened America’s position when it came to steering the course of events, and allowed reactionary forces to take control and eventually trigger off the rise of Islamic State. If the president in office is more sensible, a stronger American presence might cement the gains of the surge and result in a stronger and more stable Iraq.
Regardless of that, there would be no excuse for Putin taking a whole country. Everyone along the border with Russia, or feared Russian strength, would start rearming. The Germans rush to make use of Russian natural gas, for example, would probably be short-circuited; I’d expect Ukraine to make a play for NATO and EU membership before Russia could turn its attention to them. I don’t know if the Germans could bolster their military to any great extent at that point, but it would be much harder for them to become dependent on Russia. (Although it was clearly unwise in OTL, so I’m unsure).
Another possibility is that Ukraine would sell out for the best it could get. If Russia appears much stronger, they might decide there is no hope of victory and accept some degree of subordination within the Russian Federation. If that happened, Russian troops would suddenly be on the Polish border, a serious threat to NATO. How would America and Europe respond to that?
The Russians would have to be insane to risk striking Poland, although they would want to do it. A formidable Polish Army, with or without NATO/US support, would be a clear threat to Russian interests. Nations don’t like constraints. Would Putin gamble he could invade and crush Poland before the US could intervene? It seems unlikely, but so did his gamble in Ukraine. The United States would still have most of its easy-deployable military forces in the Middle East, and the Europeans had cut their armies to the bone. Putin might just think he could get away with it.
Lots of interesting (and horrific) ways this could go.
Chris