My personal interpretation of the line:
"Jack, I swear... [I will always love you, and I will never forget you.]"
Mine, too. Something open-ended but overflowing with emotion. I don't think Ennis was mentally filling in any specific words after the "I swear" and just not voicing them. I think that his "Jack, I swear" nebulously represents the huge jumble of feelings -- love, grief, regret, affection, sorrow, loneliness, devotion, despair, etc. -- that constantly threaten to overwhelm him at this point.
It's kind of like "Jack, I will always love you" -- except saying "Jack, I swear" is way better, because 1) we have never heard the phrase before, so it doesn't sound the least bit trite and 2) it encompasses more potential meanings than simple love and 3) it draws viewers in, like so many things in this movie, and forces them to figure out for themselves what's going on in Ennis' mind, which makes us more involved and empathetic.