According to current law they don't owe anything more [1]. That doesn't mean that other people have to like it, even monkeys get upset when they feel like things aren't fair [2] . I think it is an open question about how companies like google will keep up good relations with their neighbors in the longer term.
So all these issues would go away if Google paid roughly the same effective corporate tax rate as other companies?
I think the real issue is that people feel especially entitled to a share of Google and its employee's earnings, which still makes no sense to me. If by "neighbors" you mean the local community, then I don't see why Google owes anything in particular to the local community. They already pay local, state and federal taxes. If a rich guy moved next door to me, I wouldn't expect him to contribute more to the local community just because he happened to live near me. Furthermore, trying to extract rent from companies will fail in the long run since they are free to move within the US.
[1] http://www.bloomberg.com/news/2010-10-21/google-2-4-rate-sho... [2] http://dailypicksandflicks.com/2012/04/26/monkey-fairness-ex...