When is it appropriate to apologize via email?

Over the last few weeks I have thought a lot about the words I am sorry.  I don't know about you all but I say the words I am sorry a lot.  If someone is having a bad day, someone dies, loses a pet, any number of reasons will spark an I am sorry from me but am I really sorry.  Most of the time what I really mean is I hate that ___________ is happening to the person.  Sometimes, I mean, I know how bad that hurts and others I mean absolutely nothing other than it is a polite thing to say when in truth I really do not care at all.  Saying I am sorry literally means that the I has some ownership in the event that has occurred.  Am I really sorry that a loved one has died, not unless I was somehow responsible for the death.  Anyway, this thought brought me around to when are we really sorry and how should one apologize? 
   Technology is good thing in the world we live in but it is also a curse.  People are immediately accessible via email and cell phones.  Facebook puts us in touch with hundreds of folks we would not have access to without technology. Is it ever okay to apologize via email?  By this I mean a real apology, not merely a substitue for hating something they are going through, a real heartfelt apology where you take ownership of the wrong you have done or the pain you have caused.  I recently received what is supposed to be a sincere apology via email though the words I am sorry were absent....and there was a lot of explanation why the event happened to begin with and lot of talk about guilt.  I am perplexed by this for some reason now.  It has never occurred to me to question the method in which the apology was delivered.  I write about this now because I am keenly aware that the manner in which the email is read will play a significant role in whether the apology or resemblance of an apology is accepted. 
    It also brings up, for me anyway, the question of questions.  What if there are parts of the email apology that are unclear to the reader?  Is it appropriate to call or email a response inquiring for more clarification or does proper ettiquette say that we should merely accept the apology and forget about it?  This also brings up for me the question of is it ever appropriate not to accept an apology and what would it look like if one refused to accept an apology?  The three words I am sorry are powerful and loaded with a lot emotion or they are void of any kind of meaning whatsoever and encompass a lot of gray area in between.  So, I ask you to share your thoughts, if you have them on apologies.  I would love to hear from you.