Are both the sentences correct?
I don't care whether you're married or single.
I don't care if you're married or single.
When to use whether over if?
1) We use whether and not if after prepositions:
*Later I argued with the doctor about whether I had hit my head, since I couldn’t remember feeling it.*
Not: *Later I argued with the doctor about if I had hit my head …*
*The police seemed mainly interested in whether there were any locks on the windows.*
Not: *The police seemed mainly interested in if there were any locks …*
2) We use whether, not if, before to-infinitives:
*I don’t know whether to buy* the blue one or the red one.
3) We use whether, not if, after prepositions:
[talking about a trip to Australia for a year]
We’re not interested in whether we get great jobs and that kind of thing, we just want to have a good time.
4) We use whether, not if, directly before or not:
*Can you tell me whether or not you’re interested in the job.*
(https://dictionary.cambridge.org/grammar/british-grammar/if-or-whether)
I your case I would use whether or not, e.g.: I do not care whether she is married or not.