I posted a similar question asking if there's a way for Arduino to send a tone to a bluetooth headset. I got it working by having a Bluno on both ends using SPP. I recently learned about the different bluetooth profiles and I suppose if I want to be able to send audio directly to a bluetooth headset or earbuds I would need a board that has HSP (headset profile) and/or A2DP (Advance Audio Distribution Profile). If I had a board with those profiles, and also had an audio file I wanted to send, would I be able to pair that board with a bluetooth headset or earbuds and play that audio file?
I have only used SPP protocol Bluetooth modules with Arduino, however I think that the answer to your question is yes you can do it if you have the correct module that uses the profiles mentioned.
Such modules are likely to be much more expensive and harder to find than the commonly available SPP type.
I haven't been able to find an arduino board with those profiles, but I was looking into the RN-52 from Sparkfun. That looks promising, but I'm still looking. I figured the price would be a bit higher than the SPP types. The breakout for the RN-52 is close to $50.
Than breakout board looks promising.
Sparkfun give lots of useful information about it in their hook-up guide.
Maybe someone else with experience of similar modules can help further.