Also Tim, when you select "iPhone" as a digital media preset, InDesign Defaults to a low-res iPhone 4 resolution of 320 x 480. Then, when you go to export it and you select iPhone 6 resolution, it IS a different aspect ratio.
When initially setting up the document in which you are going to develop the content, override the preset with the phone dimensions you are targeting, and make sure you use those same dimensions when you export.
At Bates, we typically will develop at an iPhone 4-retina resolution of 640 x 480. This will resolve properly on a 4S and get letter boxed (across the top and bottom) on the 5 and the 6 (it actually resolves nicely on an iPad mini too!). However, if you design for the 5 or the 6, it will get letter boxed vertically on the smaller screens which just looks very unnatural!
Just make sure whatever size you set up the document with, is the same dimensions (or at least aspect ratio) that you use when you export - They can be overridden at each point.
Well, the content must conform to the aspect ratio of the target device (Phone or Tablet). If you don't want to resize all of your content, you could letter box it to get it to the proper aspect ratio, or adjust all of the layouts, or redevelop the content as responsive HTML. As it sounds like you would like a quick fix, I would recommend letter boxing to get to the proper aspect ratio.. Hope this helps...