First, Apple may wish to restrict applications, but in practice, all versions of the iPhone can be jailbroken: you can then upload arbitrary machine executables. Whether the phone vendor likes or dislikes that you jailbreak the phone is immaterial to evaluating its computing power. But, for the same of the argument, let's add the hypothesis that the iPhone is not jailbroken, and more generally that you only use the iPhone in ways that are approved by Apple.
Whether you can load your code without Apple's approval or not is not relevant. The iPhone is Turing complete if there is some way of programming every computable function and of encoding arbitrarily long inputs and outputs. Turing completeness doesn't say that the encoding has to be what you consider the most natural, which is to install arbitrary machine code.
There is a killer argument which shows that the iPhone, like any other computer, is not Turing-complete: it only has a finite amount of memory. Therefore, the class of computing power is that of a finite automaton, no more.
If the iPhone was allowed to rely on an external storage medium that allowed to store an arbitrarily large amount of data (finite, but whose length is not bounded by the input size), and if network access was allowed, then it would be Turing-complete. One way would be to browse a website containing a Turing machine emulator or other Turing-complete computation mechanism, with input being defined as typing in a form and output being defined as changing the content of the page.
There are other ways of encoding Turing machines that don't require network access or non-default apps to begin with. You would need an iPhone augmented with an infinite tape to pull it off, though, so such an encoding would have to specify how the tape is used in addition to the use of actual iPhone features.
nteract with all hardware devices, but as far as pure computing power is concerned, it is sufficient to perform any calculation.