I\'m not sure what the correct size should be.
Many sites seem to repeat that the apple-touch-icon should be 57x57 pixels but cite a broken link as their source.
I can not cite a source for those sizes because the official reference is under lock and key of ADC.
However, many of the non-NDA sites have the tutorials on how to create the icons. For example here:
You don't have to bother for correct size any more. If you have itune artwork file (i.e. file of 1024*1024 size) of your icon, then I have created this application which will provide you all the icons based on information provided here. Get the application from here, and follow the instructions in readme file to create all the required icons for iOS application.
Updated list October 2014, iOS8
List for iPhone and iPad with and without retina
<link rel="apple-touch-icon" href="touch-icon-iphone.png">
<link rel="apple-touch-icon" sizes="76x76" href="touch-icon-ipad.png">
<link rel="apple-touch-icon" sizes="120x120" href="touch-icon-iphone-retina.png">
<link rel="apple-touch-icon" sizes="152x152" href="touch-icon-ipad-retina.png">
<link rel="apple-touch-icon" sizes="180x180" href="touch-icon-iphone-6-plus.png">
Update 2014 iOS 8:
For iOS 8 and Iphone 6 plus
<link rel="apple-touch-icon" sizes="180x180" href="touch-icon-iphone-6-plus.png">
Iphone 6 uses the same 120 x 120 px image as iphone 4 and 5 the rest is the same as for iOS 7
Update 2013 iOS7:
For iOS 7 the recommended resolutions changed:
The other resolution are still the same
I don't think there is a "correct size". Since the iPhone really is running OSX, the icon rendering system is pretty robust. As long as you give it a high-quality image with the right aspect ratio and a resolution at least as high as the actual output will be, the OS will downscale very cleanly. My site uses a 158x158 and the icon looks pixel-perfect on the iPhone screen.
As part of their Safari Web Content Guide, Apple actually has a publicly available page called "Specifying a Webpage Icon for Web Clip" (Javascript required) which covers all of the resolutions and their implementation.
The official size is 57x57. I would recommend using the exact size simply due to the fact that it takes less memory when loaded (unless Apple caches the scaled representation). With that said, Rex is right that any square size will work