Apple is unlikely to face anti-trust action, either in Europe or the United States, so long as it does not engage in illegal behavior.
Monopolies are generally not per se illegal. Conversely, if you had 5 competitors, each with a 20% marketshare, it would be illegal if they conspired or otherwise colluded on such things as pricing or distribution.
Secondly, in no jurisdiction is it considered "anti-competitive" or "illegal" to assert one's IP rights in Court. (This is one of the fundamental errors made by the cabal of Applehaters who seem to haunt every Samsung thread on the forum.) If Apple made a habit of filing lawsuits that were adjudged to be frivolous that would be different. But "frivolous" has a fairly steep legal standard to prove.
Apple's control over strategic components is, of all the things the OP mentioned, the one that might, possibly, lead to Anti-Trust or Anti-Competitive actions. Apple is certainly allowed to own stakes in, and make investments in, companies that supply things like batteries, screens, memory and the like. Where it could potentially lead to problems is if it could be shown that Apple had acted to restrict or withhold some items from competitors. If, for example, Apple bought the company that made Tegra processors and then shut it down, that would be an example of "anti-competitive" behavior. Simply getting a good price on screens or memory because they buy in very large quantity is most certainly NOT "anti-competitive."
Both the US Justice Department and the EU Competition Commission look at factors such as the overall market firms operate in. While Apple certainly holds a very large share of the Tablet-computing market, consumers of such devices have many, many other options that provide a similar, if not identical function. A consumer may instead buy a laptop, or use a smartphone - neither of which are markets that Apple holds a dominant market position in.
Lastly, I would make the observation that Apple can afford, and generally hires, lawyers who are the best in the world at what they do. Such counsel, while expensive, will generally advise their prominent clients to avoid actions that have even the suggestion of anti-competitive behavior.For example Apple has, for the most part, avoided growing by means of acquisitions. If Apple had achieved an 80% marketshare by buying RIM, Samsung, and Motorola's tablet businesses - they would face certain regulatory opposition. Achieving 80% marketshare by making the best tablet is a whole different situation - that is simply consumers choosing a better product.
