If it comes so far (which I really doubt) as them simply ignoring it, their European Operations can be fined (directly through their bank accounts if need be) or their European employees can be jailed.

They can also be disallowed to do business with European consumers and companies.

In short: "The fines will continue until compliance improves. Get in line or get out of the market".

They don't have much political leverage, given how few people they employ and how little taxes they pay over here.

Their standing in the US isn't the best either so it's doubtful whether America is willing to invest much political capital over something she herself is considering.

They won't simply ignore it, but they will do the minimum amount they can possibly get away with, which will amount to effectively ignoring it.

The European Union is a huge market. If Google or Apple doesn't comply then they say goodbye to those profits.

It’s EU law, not US law. US cannot sign off on EU law.

considering the status of both companies as basically a duopoly, it will be hard for the US to object about unfair trade rules.