'I don't care' can be taken with a lot of things. It can also be as 'i don't care about your past because i decided to love you as you are now'. Yes he has the right to know but he decided against it because it wouldn't change the present and let sleeping dogs be.
I agree that he has to reassure his lover but you have to take in mind they're just starting their relationship. It takes TIME to properly build trust and confidence in your lover. Being 100% confident in your relationship from the get-go is just fiction.
oh wow, I can't agree more with your interpretation
love means accepting, but it's accepting after you've seen all the sides of your partner, not just accepting when you don't know a single thing!
love is about intimacy and compromise - you have every right to know everything about your lover, even his bad sides; and still forgive him after knowing all of those.
how can you actually say 'i don't care' when you really love him? he's afraid of you finding out his bad sides and leaving you, because he doesn't believe that your love is strong enough to forgive him. so your "i don't care" and ignoring it would just prove him right.