It has nothing to do with NATO, it is just another lie invented as excuse to invade. Here they still have it on the kremlin website:
"I am absolutely convinced that Ukraine will not shy away from the processes of expanding interaction with NATO and the Western allies as a whole. Ukraine has its own relations with NATO; there is the Ukraine-NATO Council. At the end of the day the decision is to be taken by NATO and Ukraine. It is a matter for those two partners."
And? That statement occurred 4 or 5 months before the USA had withdrawn from the ABMT.
I don't know enough about Russia's changing attitudes towards the USA: how much of it was hope that they could work things out together to jointly oppress the rest of the world; how much of it was biding time in a weakened state before they could mount a plausible defence against the USA.
But it does seem probable that the prospect of being left without the possibility of firing nuclear weapons at the US while staging bases are set up ever close to Russia would be undesirable to Russia.
Do not take from this that I am defending Russia's actions. I have no interest in either a US nor a Russian Imperium and think that the Ukranian right to manage their own affairs without being attacked is paramount. The same way that I believe that Venezuela and Iran have a right not to be attacked.
"I am absolutely convinced that Ukraine will not shy away from the processes of expanding interaction with NATO and the Western allies as a whole. Ukraine has its own relations with NATO; there is the Ukraine-NATO Council. At the end of the day the decision is to be taken by NATO and Ukraine. It is a matter for those two partners."
http://en.kremlin.ru/events/president/transcripts/21598