Hi,
1. Sorry my friend, I can't talk to you now. I'm learning by heart some new words.
- Is this natural to say?
Where I live, I don't commonly hear people calling their friends 'My friend'.
I'd say Sorry, I can't talk to you right now. I'm learning some new words.
If you really want to use 'by heart', it's more natural to say it in this order: I'm learning some new words by heart.
Best wishes, Clive