Can the insurance company of the person at fault refuse to pay for the damages if their client claims he or she is not guilty?

In a case of word vs word and no witness or other proof on either side, the insurance company has a responsibility to their insured and must take their version of the accident, in which case, they would deny the claim to the other party in the accident.

what if the at fault driver admits her fault but insists to settle outside of her insurance, but meanwhile refuses to pay for the damages