First, the WTO doesn't really have any direct power. They merely act to provide some common framework for international agreements, and a place to settle disputes. But if the participants of the dispute do not settle, the WTO cannot do anything. So saying "the subsidies are illegal under the WTO" does not really make sense; they violate the rules, but that's not the same as being illegal. It's a voluntary club. It should also be noted that both the US and the EU are by far the biggest complaintants in the WTO - there's some criticism the WTO server mostly to maintain the position of the strongest members at the expense of everyone else (as seen in the distribution of WTO-approved tariffs between the different countries, which are sometimes eerily similar to colonial-like relations, though the disparity is much smaller than old-school imperialism).
The free-market answer is simple - reap the massive profits you get from another country subsidizing its services through money raised from taxation (or printing money). Don't forget that Google et al. aren't the only interested party here; of course they're going to object to any competition (corporations aren't exactly friends of free markets). But on average, assuming the subsidies actually help estabilish similar services in Europe, the services are going to be cheaper and better. Of course, it comes at a cost to the EU taxpayers, so this is more true for people outside of the EU than inside.
Of course, the assumption that subsidies will help is a huge one. Subsidies generally tend to decrease efficiency, and lead to poor investments (after all, if the investment wasn't poor, it would be made without any subsidies). Even if some useful service comes out of this, which is doubtful, most of the money is going to be wasted, again increasing the real cost to people inside the EU. I've worked on software subsidised by the EU, and it's no different from most EU subsidies - just a way for a few companies to get great profit margins while doing nothing of particular value.
The best the US can do is wait. If the project succeeds, US citizens are going to benefit thanks to the increased competition on the market. Some may be hurt (e.g. losing their job in Google), but that's always true, regardless of government intervention; it's not like employees of such companies will have trouble finding new jobs even if they had to leave. More likely, the EU wastes a lot of money to create nothing of real value, and nothing really changes (except for the money now missing from the pockets of taxpayers within the EU).
Of course, this is just the market picture. There are more interests to the US government than ensuring the prosperity of its citizens. Individual politicians have their own goals and means. Individual companies and corporations will try to put pressure on the politicians to maintain their dominant position. The security services are surely happy to have almost unlimited access to personal data of a significant chunk of the world, in and out of the US. But that's nothing new, and of course, the whole thing started due to the same on the part of the EU.