Is it possible to create iOS apps with Mathematica?

Here are my few cents to have some food for thought. Can't fit this in a comment.

(1) I'd suggest indeed to take a look at the Wolfram Cloud preview and see that all tools to hook up computations to mobile apps will be there. Such functionality like listed below is specifically dedicated to that - for both: API hooks to your own mobile interfaces or complete start-to-end interface development with Wolfram Language (WL):

• Cloud Functions & Deployment
• Creating Instant APIs

• Creating Form Interfaces & Apps

  Please do not bombard with questions about Wolfram Cloud - whatever is known before the release is all in documentation I linked to. For the rest please wait for the release.

(2) Consider efficient design & prototyping. In this respect look at some famous cases. "Disney Animated" was selected by Apple as the Best iPad App of 2013 and it was designed with WL by Touch Press company and detailed here by founder Theodore Gray. Read that article to see the workflow. Quoting:

In order to create this effect in real time on an iPad, we of course needed to write highly optimized C and OpenGL code. But before we could do that, we needed to refine the algorithm and parameters through a series of prototypes and meetings with the visual effects supervisors at Disney, to be sure our implementation maintained the spirit of the original, even though it could manage only about a tenth as many snow particles. That prototyping would have been very painful to do in C code, but by building the prototype in Mathematica, I was able to make adjustments on the spot, and in one case almost completely rewrite the algorithm overnight, then take it back to them the next morning in much improved form. ~ Theodore Gray

(3) Large scale development example is Wolfram|Alpha where heavy computations and NLP run in WL server-side and together with data are hooked-up to Wolfram|Alpha App via API. Also think Siri which is an "iOS app" ;-) and is linked to Wolfram|Alpha.

(4) Don't forget about C code generation and tight integration with other systems - can come very handy during development:

• C Code Generation User Guide
• Systems Interfaces & Deployment

The answer to your question is given in the comments. In condensed form the answer boils down to: for version of Mathematica up to V.9.0.1, it can not be done. However, we have hopes that some sort of CDF player will available for iOS in the near future, perhaps as part of the shortly expected V10 release.

Wolfram released Wolfram Player for iOS this fall. The free player can load CDF files from other platforms. Users with a paid Wolfram Cloud subscription have the full functionality of the CDF -- that primarily means CDFs that use Manipulate() to provide interactivity. Users can also get that functionality with a $9.99 in-app purchase.