From the European side of things i believe that this missile shield was not popular among citizens plus it was a potential threat to our relations with Russia .
I remember some tv reportage from Prague claiming that 99% of the citizens were against the missiles while 100% of the political parties were supportive.
What about the loss of trust between Western Europe and the US?
I would prefer stronger rather than weaker transatlantic ties, *if* we can trust America to behave in a reasonably sane manner. Trouble is, GWB made a rather a big dent in that trust. Obama seems sane enough, for sure, but who's to say they don't elect Palin/Tancredo in 2012 or 2016? If *that* happens, I would very much prefer not to be all that dependent on the US for security matters; i.e., strong transatlantic ties are just fine as long as we keep a big pair of scissors handy… just in case.
Western Europe and the Americas are tied to the same boat , there can be trade wars , misunderstandings or even short term "cold" relationships but open hostility or long term contrasts are unlikely.
Same goes to Russian-EU relationships, Russians can trash Georgia or whoever they want for as long as both them and EU are making money .
Just my humble opinion
Due to the missile shield? Negligible compared to the fallout from other issues.
Not much trust left to lose after "axis of weasels" and "either you are with us or against us". The situation was already so bad (for trust between what after all are allies) that the missile shield was unimportant.