Is
Nobody could survive on its own
grammatically correct, or is it
Nobody could survive on his own
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Nobody could survive on its own
grammatically correct, or is it
Nobody could survive on his own
People aren't objects, so you shouldn't use "it's", you should use "their".
If I were you, I would say:
Nobody could survive on their own
You should use "their" instead of "it's", because "it's" is for objects, whereas "their" is more towards the meaning of "for people".
P.S. OP said that he uses "could", I was thinking the OP should use "can", but the OP said in the comments that he uses "could".
Since 'nobody' is a personal pronoun, it refers to a person. No person can be referred to as 'it', so 'its' won't work well there.
Were the sentence 'Nothing could survive on its own', then the its fits.
Nobody could survive on his/her/their own is acceptable.