You're on the right lines I think, keep it simple:
"I'm sorry I looked at your personal stuuf. I can understand why you're angry with me for doing it....it was disrespectful and I shouldn't have done it. I'm Sorry".
IMO...You need to have in mind the purpose of doing the right thing - ie owning up to something you shouldn't have done and then exit the conversation with some dignity and by being in the right i.e. not justifying or defending yourself - you were in the wrong on this one even though she shouldn't be doing what she's doing.
Best - GFI
Me: 40ish W: 40ish Together: 20 ish years Married: 10ish Years