First off, after the mission "No Russian", wouldn't the Russian authorities have launched a massive investigation, which would involve identifying the three other gunmen? Which would, in turn, reveal that the leader was a known terrorist and madman.
Two, since when is it considered sane to declare war on a country for the actions of one crazy-person (who wasn't actually crazy)? Some may cite the assassination of Archduke Ferdinand and WWI, but I (preemptively) counter that there were already factors and men/idiots who were pushing for war and the assassination was just the straw that broke the camel's back. As far as we (and the game) know, war is declared for the sole reason of an American being involved in the slaughter.
Three, the entire invasion force is shown to be attacking from the Pacific and the Atlantic. The Pacific? Yeah, I can buy that, since Russia has direct access to the Pacific Ocean. But the Atlantic? No way. That would mean that Russian forces had to either a)launch from the Barents Sea and make a long sea trip, b)take a massive road trip through Europe, or c)fly massive armadas over Europe, all of which involve (illegally) passing through the borders/airspace of almost every major European power.
Four, how do you launch a massive, full scale invasion force in the digital age, which has almost-instantaneous-communication and satellite surveillance, without anyone noticing until the last minute?
And finally, the coup-de-grace, the most glaring hole the Russian invasion in MW2 has. The entire invasion force: all the men, the equipment, the armor, the aircraft, the supplies and ammunition. All of that was gathered, prepped, organized, and mobilized, within the span of three days.
No sense at all.
Don't get me wrong. I don't hate MW2. It's a solid and enjoyable shooter. It just didn't think things through.