Apologize first - ask questions later.
ExcuseMe?
ApologizeFirst may be more acceptable than ApologizeUnconditionally, which can be viewed as a sign of lack in sincerity. See more below in section "Half a glass"
Do it with an apologetic smile, and mean it that way. You want to be friendly, you have something important to communicate, but you are unsure whether the other side is ready to hear you out. You want to OnlySayThingsThatCanBeHeard.
If you initiate the communication, you may need to apologize for interrupting the other person, even if you both may agree later on that you "did the right thing".
Whenever people interact, they may offend one another. Such offense can prevent further positive interaction.
It is contributory to dialogue that the communications between parties be of the type "No offense meant" and "No Offense taken". Rather than apologizes for how something is taken by the other party one must CommunicateFirst?. To communicate, one must first establish a basis dependent upon mutual interests and cooperation. For one to communicate, meanings, not words must be exchanged. It is true that people may hear you, but not understand you. This is not necessarily because they lack the ability to understand your words, as much as they fail to perceive what you mean by your words (or your silence). In most cases, it doesn't really matter whether the offense was intentional, and it doesn't matter who is "right" and who is "wrong". All that matters is that the ill feelings must be prevented or minimized, so that positive interaction can resume.
The longer the bad feelings exist, the more difficult it will be to eliminate them.
Therefore,
Consider ApologizeFirst, with sincerity.
Half a glass
NothingIsWhatItSeems?, including Apologies. Further to TrustButVerify UserStory ShatteredGlass, some people are pointing out to the "possible" lack of sincerity of a previous liar whose motive appeared to seek fame and fortune (not "acceptance" as explained by the main character). See http://slate.msn.com/id/2091015/
Contrast with NeverApologize