It really depends. Apologies can be a tricky business, but tossing them out can help bring the temperature down to move forward. Sometimes you have to read between the lines to determine whether it's useful or not, or what exactly they want an apology for (it's not always what you'd think at first).
Looking at your example in the end, I'd take the following tack - "I'm sorry, I didn't mean to offend you. When I'm here, I try to stay really focused on what I need to do. If I'm not super responsive or don't notice you right away, it's because I'm concentrating on something else. It doesn't mean that I am deliberately ignoring you, or think badly of you. [If you can muster it/can honestly say something nice about them, do it. It helps. Has to be genuine though.] I hope you understand."
Might calm things down, might not. But I've had folks respond to this approach before (when they just want to feel acknowledged, which is where most 'y u ignoring me' reactions come from).