Earlier quoted context omitted.
There is no way they would make enough from iPhones/iPads to pay for the costs of building and running a competitive web search engine. (At least not one built like Google and Bing are) There is a reason they take a cut of profits from App developers; it's to help pay for the cost of running the App store.
They're generating profits at a rate of $200 Million per day [sic]. It's fair to say they make enough to pay for the costs of almost anything, including this. You've identified the one requirement they can in fact satisfy; the hard part is... everything else. Edit: Clarity
"For the full fiscal year, Apple generated $182.8 billion in sales, a new record for the firm, with $38.5 billion in net income."
$105MM, not $200MM.